Color Symmetry and Quark Confinement

Home , Color confinement

COLOR SYMMETRY AND QUARK CONFINEMENT

Proceedings of the

TWELFTH�ENCONTRE DE MORIOND. I 2�.

Flaine - Haute-Savoie (France) March 6-March 18, 1977

VOL III

COLOR SYMMETRY AND QUARK CONFINEMENT

edited by

TRAN THANH VAN

SPONSORED BY

INSTITUT NATIONAL DE PHYSIQUE NUCLEAIRE • ET DE PHYSIQUE DES PARTICULES

• COMMISSARIAT A L'ENERGIE ATOMIQUE

3 The " Color Symmetry and Quark Confinement ,, Session was organ ized by G. KANE and TRAN THANH VAN J. With the active collaboration of M. CHANOWITZ

...... ___, \.

Permanent SecrCtariat .

Rencontre de Moriond

Laboratoire de Physique Theoriquc B<."ttiment Universite de Paris-Sud 211 - ORSA Y (France) 91405 Tel. 941-73-72 - 941-73-66

4 FOREWORD

The Xllth Rencontre de Moriond has devoted a two-day session to the problem of Color Symmetry and Quark Confinement.

The talks given at that session are collected in this book with an added pedagogical introduction. It is hoped that this book will be useful to learn about color and how it can be tested experimentally. The review papers summarize the current situation at April-May 1977.

I wish to thank all the contributors for the ir efforts in making the ir papers as pedagogical as possible.

TRAN THANH VAN J.

CONTENTS

G. KANE Pedagogi.cal introduction to color calculations 9 CHANOWITZ Color and experiments M. 25 O.W. GREENBERG Unbound color 51 J.D. JACK SON Hadronic wid ths in charmonium 75 J. KRIPFGANZ Parton mode l structure from a confining theory 89 C. �IGG Dilepton production in hadron-hadron collisions and .the "factor of three" from color 93

M. SHOCRET Production of massive dimuons in proton nucleon collisions at FERMILAB 103 G. VENEZIANO The colour and flavour in l/N expansions 113 F. LOW Comments on the Pomeron 135 ' S. BRODSKY Jet production and the dynamical role of color

Y.P. YAO Gluon splitting and its implication on confinement. A non perturbative approach 165 G. TIKTOPOULOS The infrared problem in color dynamics 189

H. FRITZSCH Universality of quarks and leptons 215

* PEDAGOGICAL INTRODUCTION TO COLOR CALCULATIONS

G.L. KANE Physics Department University of Michigan Ann Arbor, Michigan 48109

ABSTRACT An introduction to calculations with color symmetry is provided for non-experts; we proceed mainly by giving examples. The observability of color symmetry, even though only colorless states might exist, is emphasized.

� Research supported by U.S. Energy Research and Development Administration.

g I. INTRODUCTION

It seems rather likely that for some time to come the most fruitful

view of hadrons will be as composites of colored quarks and gluons. While the

theory of Quantum Chromodynamics (QCD) is still developing, it is already stan

dard to find "color factors", factors due to the presence of this internal sym

metry, quoted in many amplitudes of interest. Indeed, color is most directly

observable in terms of the effect of these color factors on transition rates.

While there is nothing particularly new or mysterious about calculating the color factors - it is, in the standard theory, an exercise in conventional

SU (3) some care is required to get it right. There are few places where an

uninitiated reader can go to learn easily the necessary techniques. Consequently, it was thought that it would be useful to include an introductory pedagogical treatment as part of the present book, and that is the purpose of this chapter.

Other introductory aspects such as the history and the main tests of color are well covered in other chapters or in references given there, so we will concentrate on calculational techniques. A few other sources are mentioned in references below. References to the original literature should be traced from other chapters and from the reviews mentioned.

Fir�t some introduction to the formalism is given in Section II. Then

in Section III a number of examples are analyzed, some with simple arguments and some with formal manipulations. We will concentrate on the standard theory, where

color is an unbroken symmetry and hadrons are only in color singlet states ; other views and ways to distinguish them are well covered in the articles by Chanowitz and Greenberg in this volume, and by Greenberg and Nelson. Finally, in

Section IV some further remarks are given about the observability of color. It is worth emphasizing again that the purpose of the present notes is to give an ele mentary but fairly complete, pedagogical, introduction to the use of color sym metry in quark physics.

II. FORMALISM

We assume that quarks come in a triplet representation of an unbroken

SU (3) color symmetry, and that they interact by exchange of an octet of colored

gluons. That there are 3 colors and the symmetry is SU (3) is the standard choice to have three quarks in a baryon, and have them in the totally antisymmetric state required by statistics, and in a color singlet state. We will label the quark colors, as r, g, b (for red, green, blue). The gluons can change quark color, so a r quark can absorb or omit a gluon and become a g quark, etc

10 Some color factors can just be written down, as we will see, but for some calculations we need the Gell-Mann SU(3) A matrices, so let us recall their properties ; it is probably worthwhile to give essentially a self-contained treatment. They are just the generalization to SU(3) of the Pauli matrices for spin or isospin. Let us think in terms of isospin, with matrices

• T2 =(: -�) (I) and commutation relations

Ti T. (2) [- • __J_] 2 2

Combining the commutation relations with the anticommutator

(3)

one has the useful relation

(4) ·r. T' 1 J

Most of the useful properties of the A matrices can be deduced from the commutation relations and simple arguments based on the isospin analogy. We have :

>.k i f (5) ijk ( ;. �] 2 and

•• •A 2 d (6) \ j ijk i 6 { } \ + 3 1J

The second term in (6) has a 3 x 3 unit matrix factor understood. Combining these we find one useful relation analogous to (4), 2 • � f A d A ijk k ijk k -:;- 0ij + +

Many complicated expressions with A matrices can be handled using this equation.

The numbers f and d are the standard SU(3) constants, available in many ijk ijk textbooks. Just as for isospin, at a quark-gluon vertex there is a factor due to color A1 ab

11 where a, b are quark color labels , a, b = r, g, b or a, b = I, 2, 3. When calculating color factors we expect only internal gluons , whose color "pola rization" is sulllll1ed over. Therefore we will find factors such as

i i E Aab d i \ which need to be evaluated . The appropriate identity is

i i A E ab \d 6 ad 6 be _.!._6ab 6 cd (7) i 6 2 2 2

To check that this is correct, note the following. We can write down a few >-'s by analogy with isospin. We can take

" - (; : :) merely the Pauli matrices with a third row and column , still traceless and her mitian. Since

then for i 3 I, 2,

� I, ? In particular , note Tr>- = for i 3 and so Tr\ = for any l 2 = 2, l 2 One more A is obvious , the one wh ich is diagonal in isospin , usually called \8 Since it has unity in the II and positions and is traceless and normalized 22 2 = it is with Tr\8 ,

Now , to check eqn.(7) , we can (i) put b c and sum, giving the matrix product, (ii) put d a and sum, giving the trace.

1 2 Since each term gives and there are 8 terms we get 16, divided by for the two 2 4 factors of 1/2. The right hand side gives

9 aa cc (3) (3) - (3) 0 0 6 0 ac 0 ac = 6 4 ) 2 2 2 as expected.

I f we put a b c = d 3 , all but >- have a zero in the 33 posi- ti on so we get 8

3 4 �c"�s1 4 and (�)

3 2 6

• 2 A third check is to take a c b d Then and = = = 2 (>-: ) = I -1 so both sides should give zero . These three independent2 checks es("�ta2blrish the identity of eqn. (7) since there are only three ways to combine the indices. Finally, we assume that hadrons are color singlet states . To have nor malized wave functions, then , a meson is represented by

(8) and a baryon by

(9)

Now we proceed to various physical manifestations of color symmetry.

III. EXAMPLES First, we can write the electromagnetic current as E M ••• J + u - d d d d + 3_ r u r g g b _!_ r r + g g db b (IO) µ = 3 (� � + � �) 3 (d + ) where the dots are simi lar contributions for other flavors s, c, etc ... The nor malization of each term is required by the c��rge of each quark .

1 3 Now us calculate the ratio ler

transition is through an intermediate photon . The It has been shown that the total rate to hadrons can be calculated as if there were a transition to free quarks (which then turn into hadrons). Then y -+ qq and q

"-l:-Y�- � y color: � sc will all colors contributing equally we get a factor of Nc He can see the same result and practice with the formalism. The photon -quark vertex for quark colors a, b is given by cab and the color singlet /3 wave function is car,J so

2 so the cross section is larger by M "' 3 • If there had been Nc colors this would become M "' IN;; , a "' Nc so F gives a direct measurement of the numbers of colors as well as the presence of such a symmetry. Next let us consider the Drell-Yan process, where a qq pair from two interacting hadrons annihilate to a lepton pair via an intermediate photon. Again, we are only considering the color factor, which here is expected to be 1/3. More extensive discussion can be found in the chapter by Quigg . Let us derive this factor a couple of ways. r bb)//3 . Then when blue Consider color singlet mesons, ( r + gg + quarks annihilate we have

with the circled quarks annihilating . For green quarks, similarly, the contribu tion is

The remaining quarks are blue in one case �nd green in the second , so they do not interfere. Consequently, we square before adding. The amplitude for each color is l/3, so the rate for each color is 1/9 , so the total rate has

1/9 + 1/9 1/9 1/3 , an overall factor of 1/3 from color. + =

14 We can also view the Drell-Yan process in terws of the conventional formalism. Ignoring color we would write the proton structure function

ep F (x) x u(x) (x) d(x) (x) s( ) (x ( + u + ( + d ( x + s ) 2 [� ) -} ) +-} ) 1 where u(x) , etc ••• is the distribution of u quarks with a fraction x of the proton momentum in the scaling region .. Including color we would instead write

ep F (x) 2 and

u (x) u(x)/3 c r, g, b c = since the measured u(x) summed over color. For other color models the last equation might not hold, with possibly a varia tion depending on x or q2 • The Drell-Yan rate depends on the product of structure functions from each hadron, and the annihilation is due to one pair of colored quarks, so a factor l/N� comes into the rate. However, there are Ne more pairs than without color, so the final factor is I/Ne as above.

Now we turn to calculating color factors for several processes where it is most convenient to use the formalism of SU (3) A matrices summarized above.

First consider qq scattering by single octet gluon exchange. Consider the diagram

d •

i i M A A � ab cd 4 l i Each vertex has a factor A /2, and the virtual gluon color state is summed. The initial and final states can be thought of as having the quarks cooe from diffe

• rent hadrons, perhaps to undergo a hard scattering to large p� Then the cross section is

el: = ab d

J.. (±)(�) -- Since the SU(3) matrices are hermition, A ' , and we have summed over all ab )ba final quark colors and averaged (giving 1/3 for ,each initial quark) over all

15 initial quark colors . Since the initial quarks are in color singlet hadrons the probability of fi�ding a qua�k of given color is 1/3 c�(oaJ13)(oa,J13)=1/3oaa for mesons , E ' = l /3o , for baryons , so in averaging � /6 abc /6 a bc aa ) c( ) ( ) , over the initial state we have a fac tor 1/3 for each quark it is like spin states with an unpo larized beam E if we start with color singlet hadrons . Using eqn . 7, we get

E 9I (_!_o o - � o I o o � o a = abed 2 ad cb 6 ab cd )( 2 be ad - 6 ad c d )

2 - -(3) (3) --I (3) --I (3) -I (3) (3) 910 b. 12 12 + 36 ) 9

For a physical situation the relevant calculation is more complicated (see Feynman and Field for more discussion) . One mu st add the crossed diagram where quark b comes from the lower vertex . The size of the interference term depends on the scattering angle, and is different for different color symmetries . For a given physical process one must add the relevant diagrams with spin and color fac tors , and explicitly calculate the cross section .

Next, let us examine the effect of color on several decay rates . We ignore changes due to electromagnetic and strong coupling strengths . In all cases when we speak of the effect of the color factor , it should always be interpreted as meaning a comparison of processes , such as the effec t relative to positronium or of two different final states .

a) Consider the expe ted rate for the harmonium state n 2y , as c c c � shown , with color labels in parentheses ,

(a) y =r=(e) c (b) y

Formally, the matrix element has o / 13 for the initial wave func tion, and ( ab ) e , o at the quark-y verti es . Then ae be c

so the rate is enhanced a factor of 3 by color (comp ared to the analogous rate for parapositronium) .

16 b) Next cons ider . Here , as reviewed in detail by Chanowitz , TI0 + 2y it is we ll known that the expected enhancement due to color is 3 in amplitude , 9 in rate . But what is different from n 2y just above It is the role of c + ? PCAC and the triangle anoma ly that is different. The ase is viewed not as TI0 c a quark annihilation but as

shown here ,where the couples to the divergence of the axial current , and

three currents are coupled to the quark loop . The coupling of n° to aµAµ is ay fa tor f tively in ludes the olor via the measured dec c n , which then effec c c wave function normalization . The coupling of the axial current and two vector currents to the quark loop is normalized by the currents . Clearly with three colors there are three loops and the amp litude is enhanced by 3., the rate . by 9.

Comparison of (a) and (b) illustrates nicely that one must pay some attention to the detailed dynamical situation when working out color factors .

(c) Consider a hypothetical heavy quark b of charge -1/3 , coupled to the u quark in a weak isospin doublet (Cahn and Ellis) . The nonleptonic decay of a bu meson state to sc would proceed via a direct channel interme - diate vector boson W , as shown

::>-Y--

so the rate is enhanced by 9 . If the bu meson decays into a lepton pair such

as µvµ , there is no color factor for the final state and the enhancement is 3 (the reverse of (e+e- +hadrons)) .

(d) Alternatively , consider the hypothetical bd meson going into uu .

Then the W is a t-channel exchange ,

17 b • u V- �\ u d • I and since the coupling at each vertex is just the Kronecher delta coupling of the current , the color factor is just unity in the rate.

Finally, consider the QCD calculation for the total hadronic decay (e) + 2 gluons . Since the two gluons are assumed to have unit probabi lity of nc , nc to become hadrons , it is sufficient to just calculate the annihilation graph to two gluons. The amplitude for the figure shown for quark colors a, b, c and gluons i, j is

- a i

c r__,. _L j

0 ac 2 2 i j Now Tr !- i- is zero if i and 2 if as shown above , so "' '

. •. 2 a 'V 4 0 8 8 The complete L . � 6 .. lJ lJ lJ 12 1 11 12 3 48 amplitude is given by the sum of thi s process and the crossed diagram. In this case since c the two gluons are identical particles and the interference M 'V ij and counting of states is the same as for final photons . Thus we can take the QED patapositronium calculation and just multiply by the color factor of 2/3

(and redefine the coup ling strength) to get the expected nc decay rate . For the three gluon annihilation of relative to orthopositronium it is more com 1/! plicated to show that the counting is the same as for photons .

(f) Finally, consider a heavy lepton decay . The expected mechanism is

L-+ \\+W W + xy and one must sum over all pairs x, y that couple to the appropriate Presumably one has x, y = ev , cs , ud plus any addi W µvµ , e tional contributions from heavy quarks or leptons permitted by energy conserva tion . Since each color couples to the current independently, there is one contri bution from each lepton , plus _Ne from each quark pair. Thus the branching ratio to a single channe l is expected to be l/(2+2Nc) if all the above are allowed. For

18 the decay of the SPEAR heavy lepton presumably cs is not allowed by energy

conservation, so colors predict = 1/5 as discussec by Chanowitz. 3 1/(2+3) ,

Next we turn to some aspects of the influence of color on hadron spec troscopy. Some of these details are available in Jackson but they are (1 976) included here for completeness ; the connections to the original literature can also be traced from his lectures as well as the articles in this volume.

If there were a coupling g in the absence of color, the single 0 gluon exchange force gives an effective coupling g between q and q, for color

singlet mesons,

where the labeling is as shown. Notice

i * . : �Cl)=��=-

the order of indices (think of q as q in the other direction).

Then

2 2 2 •. g z g 2 g 0 ij 0 0 ij 6 lJ 12 12 12

For vector, octet gluons this corresponds to an attractive force, giving binding

for mesons.

For the qqq system (a baryon) we have

a d b . -·-te c�CL:

E ' E 2 abc d � e def g z g � c abc ij o o ) cf def Id ( ; )� 2 /6 2 g 0 i j A ;\ 1 0 - 0 abl: ad be ad be ae bd �-lJ ( 6 ) 24 de

i -g� A ;\j Z l: del: ij ed de ij 24 24

19 This also gives an attractive force, half the strength of the meson binding force

Thus the electromagnetic coupling strength a in the photon exchange interaction gets replaced by (4/3)as for gluon exchange in a meson and (2/3)as for gluon exchange in a baryon. Ano ther useful way to get this result is as follows (de Grand et al, 1975) . Consider quark and antiquark bound together in a color singlet state Then we must have for any [O> .

Taking a scalar product with 1' , we have for this situation q

With numerical factors 1/4 since 1'/2 goes at each vertex and 1//3 for initial and final states, this gives -4/3 as before . For the baryon

Taking scalar products with , 1' , i- in turn and combining equations Aql q2 q3 gives

2 or half of the meson result, giving the factor -2/3 from above . The qq interaction due to single gluon exchange is proportional to the scalar product . We change just seen that this gives an attractive 1-q.Aq force (i.e., a minus sign) for color singlet meson (and baryon) states. If we write

and solve for .A (following the common procedure with spin or isospin) we Aq q have

In the color singlet state i- + = and we have our result above. For q 1-qy 0 higher color states we would( need to know the eigenvalue of i.e. ( Aq + \:jY , 2 = the analogue of J J(J+I) ' but it is sufficient for our purposes to just

20 2 note that A is positive, so that forces mediated by color gluon (Aq + q ) exchange will always have the color singlet states lying lowest. Consequently, even if quarks and gluons were not confined it would make sense to have the ob served hadrons in color singlet states. Similarly, consider a color singlet hadron and ask what is the force between it and another quark Q . We can write the interaction as

M A in hadronq�q Q"\(q) and rearranging gives

M q,qL But the quantity in parentheseswill vanish for a color singlet hadron, so the existence of only qq and qqq systems can perhaps be understood dynamically, in terms of the properties of the color gluon forces. See Greenberg 's chapter for more discussion on this saturation question. Two other ways in which the color properties of the forces may affect dynamics are interesting . First, since hadrons are to be taken as color singlets, and one can get a singlet from 3 ® or 3 ® (3 ® 3) •.• , the 3 = I @ 8 = I + (3 ® 3) diquark in the baryon must transform as a as far as color forces 3 can tell. This may help clarify the dynamical role that diquark systems seem to play and the similarity of the Regge behavior of mesons and baryons . Second, it has been remarked (Nussinov, 1976) that a dynamical symmetry property of the Pomeron may arise from color. If one builds the Pomeron from multigluon exchange , the possibility of odd charge conjugation (C) exchange naturally arises in the 3-gluon contribution. But due to color this odd C part of the exchange vanishes identically. It is given by

a k d b e c -----1 ------f i f J.J ' ' M "'l: ijk \f Abe AJad abc def where the antisymmetric SU(3) numbers f project out the odd part of the ijk c gluon exchange and all indices are summed . Since i, j and e, f and b, c are all summed over we can interchange them i<+ j , c..._ b , e<-+ f. After doing so A ' the three 's are unchanged, and the quantity fijk abc'def has had three sign changes , one from each of its totally antisymmetric pieces . Thus so = M � -M M 0.

21 IV. OBSERVAB ILITY OF COLOR

To conclude we want to briefly emphasize the ways in which the exis tence of a color symmetry may be observable. While the elegance and richness of QCD may suffice to convince many theorists of its relevance or even truth, one often hears -- particularily for the unbroken, "hidden" color symmetry -- ques tions about the meaning of having a new set of degrees of freedom that are not directly observable. Because states of non-zero color may exist, is color unob servable

Of course no t. We are familiar with procedures for observing non-abe lian groups such as angular momentum or isospin from selection rules and transi- 3/2 tion rates. If TI and N could only interact in I states we would learn it from the 9/1 ratio of elastic scattering of n+p and n-p . Similarly, we have mentioned a numb er of observable consequences of color above (see especially Chanowitz 's article for additional examples). In addition to the basic need to antisymrnetrize three quarks in a baryon, these include the predicted factor of

Ne increase in R = o (e+e-)/o(e e- + the factor of Ne decrease in the T + µ+µ-), Drell-Yan cross section , a numb er of changes in decay transition rates and bran ching ratios, such as a factor N� in nonleptonic weak decays wi th s-channel intermediate vector bosons and no mo dification for t-channel ones . Other color dependent predictions are available for a number of processes involving photons , and for scaling violations in deep inelastic v, e, µ reactions (see the Chanowitz review) . Many of these test both the presence of an internal symme try and the actual degree of symme try , the number of colors. In the near future both our experimental and our theoretical understanding of these tests will become clear , and the experimental status of color symme try will be settled by several independent measurements .

ACKNOWLEDGMENTS

I have been helped in writing these notes by suggestions from and conversations with Howard Haber , Martin Einhorn and Robert Cahn.

22 REFERENCES

For reviews, including discussion of models where color is not "hidden" , see the talks of Chanowitz and Greenberg in this volume, and F. Close , 1975 Zakopane School, to be published in Acta Physica Polonica.

O.W. Green�rg and C.A. Nelson, "Color Models of Hadrons" , Univ. Md ., Preprint 77-047 .

The second of these gives especially detailed historical references.

Other helpful treatments are

H.J. Lipkin, Lecture in Topical Conference on Deep Inelastic Scattering, SLAC, 1973.

J.D. Jackson, Lectures on the New Particle, Proceedings of the Summer Institute on Particle Physics, SLAC, 1976.

Some papers referred to in the text , and not covered in the above reviews, are R.N. Cahn and S.D. Ellis, Univ. Mich., Preprint 76-45.

R.P. Feynman and R.n. Field, Caltech preprint 68-565 .

T. De Grand et al , Phys. Rev . !!_!2, (1975) , 2060.

S. Nussinov, Phys. Rev . �. (1976) , 246.

* COLOR AND EXPERIMENT

Michael Chanowitz Lawrence Berkeleys. Laboratory University of California Berkeley, California 94720, U.S.A.

ABSTRACT

After a brief review of the color hypothesis and the motivations for its introduction, we discuss the experimental tests. We assillne colored states have not been produced at present energies and we discuss only experimental tests which apply below the color threshold, when color is a "hidden symmetry." Some of these tests offer the possibility of distinguishing between quark models with fractional and integral quark charges.

25 I. INTRODUCTION

In recent years most theorists concerned with the quark hypothesis have become increasingly convinced that it is necessary to endow the quarks with a new property as color. Color has had little impact on experimental physics known and experimentalists seem to be more skeptical of the idea than most of their theoretical colleagues. This skepticism is natural and well founded for at least two reasons. The first reason is historical: color is introduced as a conceptual "fudge factor" which seems to save the quark hypothesis by compli cating it. If it turns out that quarks are the aether of hadronic physics, then color is the aether drag hypothesis. The second reason is related to an essential property of color SU(3): it is a hidden symmetry, hidden either forever or until very high (unspecified ) energies are attained. This limits the variety of experimental consequences which can test the validity of the color hypothesis.

At the same time it is clear that we must take the color hypothesis seriously and pursue the experimental and theoretical consequences as far as we can. Color is not just an artificial "fudge factor" but provides a very elegant basis for quark dynamics. More importantly it seems to be a necessary ingre dient of the quark model, which describes a great range of data from low-energy spectroscopy to high-energy dynamics. Tests of the color hypothesis are there fore a crucial component of the effort to evaluate the validity of the quark model.

The "hidden" nature of color poses a challenge to theory and experiment.

The experimental challenge is to perform the small number of experiments which test the color idea; almost all involve formidable technical difficulties. The challenge to theorists is to enlarge ·the list of experimental consequences which can clearly test the color idea.

In this talk I will sketch two versions of the color hypothesis, briefly review the motivation for the introduction of color, and then go through the short list of consequences which can be tested experimentally. I will

26 concentrate on the energy domain in which color is a hidden symmetry; if color degrees of freedom are ever actually excited there will be many directly observable consequences .

II . TWO VERSIONS OF THE COLOR HYPJTHESIS

The two versions of the color hypothesis under discussion each require a set of nine quarks f where R, Y

1 + + (1 ) -{3 Ordinary hadrons are therefcxe colorless "white" blends of the three colors; it is in this sense that color SU(3) is a hidden symmetry.

The versions differ in the electromagnetic charges assigned to the quarks, which are exhibited in the electromagnetic current . In the version with 1 2 fractional charges, ' it is

fractional Je.m.

(2)

. J (8, 1)

The subscript (8,1) denotes the fact that this current is an octet of ordinary fractional SU(3 ) and a singlet of color su(3). So cannot transform a white Je.m. (color singlet ) state into a colored (nonsinglet ) state. 3 In the second version the quarks have integral charge: two are positive, two are negative, and five are neutral . The current is

integral (3) Je.m. +

27 With a judicious application of the mathematical theorem

x + y y = x (4) this can be written as

integral Je.m. (5 ) fractional Je.m. where

(6)

tional n ._.B If we begin with J!�:� = J(S,l) and interchange p <-'>Y, , then we obtain the new piece J ' which is therefore a singlet of � ._.R (l, S) ordinary SU(3 ) and an octet of color su(3). integral can transform a white state into Because of this new piece, Je.m . a colored state. When the threshold for the production of colored states is reached, then according to parton and li�t-cone ideas J (l, S) should cause + - a(e e ---.hadrons) to increase by a factor of two and for isoscalar targets the electromagnetic deep-inelastic structure functions and should vw2 w1 increase by nearly a factor of two (i.e., by if we neglect quarks ). 9/5 � Since no such dramatic increase is observed in the deep-inelastic muon scat 4 tering experiment at FNAL, it seems to me unlikely that J (l,S) is the agent - of any of the present commotion in e +e annihilation . An alternative point of view will be presented in the talk of Greenberg . He will discuss the possibility that we have already passed the color threshold + - but that the extra contributions to e e --;hadrons and to are suppressed vw2 by a "color-quenching" mechanism that arises in a gauge theory realization of the integral charge model. For the remainder of my talk I will assume that the color threshold lies above the energies which have been probed experimentally.

28 Consequently in this talk I will concentrate on the problem of experimentally inve stigating color when we are below the color threshold. In this regime, cannot contribute to processes involving a single electro J(l,S ) - magnetic current (such as e+e annihilation and electroproduction in the lowest order of Q.E.D.) and the two versions of color are indistinguishable from one another (though in certain processes they may be distinguished from models with no color--see Section IV). However, there are a few processes, involving two electromagnetic currents, which offer the possibility of distinguishing between the two versions of the color hypothesis even though we are below the threshold for the production of color . These are among the experimental processes discussed in Section IV.

III . MOTIVATION

Now I want to review very briefly (and unhistorically) the principal motivations for the introduction of color . At the same time I would like to convince you that color is not just an ugly complication without redeeming features but that it provides an elegant and natural possible setting for quark dynamics.

A. The Statistics Problem

Within months of the original quark proposal by Gell-Mann and Zweig, 1 Greenberg ob served that the classification of the 8 and 10 of baryons (the 56 of SU(6)) as s-wave bound states of three quarks requires the introduction of a new threefold degree of freedom (which he called parastatistics of order three and which for our purpose is essentially the three color model with fractional charges ). The point is that three quarks the 56 representation of SU(6 ) in are symmetric under interchange of spin, spatial, and ordinary SU(3 ) degrees of freedom . This violates the Fermi statistics required of spin } quarks . For instance, consider the quark model assignment of the 6::=3/2 with I = S p where the arrows denote the spin z z + s ,J!', p1'), 2. 6 z =3/2 , 1 polarization s Greenberg's- idea is based on the group theoretical �. z I = +++ ) - I"' 2.

29 fact that the singlet configuration of three members of an SU(3) triplet is the totally antisymmetric combination . Thus if we endow quarks with a color su(3) degree of freedom and require that be in a singlet of color 6++s / z�3 2 su(3), then

+ +

(7) and the statistics problem is automatically resolved .

B. The Lifetime of the Neutral Pion

This subsection also begins with a problem: Sutherland and Veltman showed using current algebra and PCAC that in the limit of zero pion mass,

is forbidden.5 Taking the corrections to PCAC in the amplitude to be TIO yy 2 �o(m /l Ge'1-), this implies that for the real pion 0 � should be TI r(TI yy) 4 �(m /l Gev) � l0-3 times smaller than we would naively estimate and than it TI 6 is actually measured to be . Adler, Bell, and Jackiw later showed that this argument omitted a contribution ("the anomaly") due to the basic triangle diagram, and that this contribution allows 0 to proceed in the zero TI � yy pion mass limit . This technical argument acquired great significance when

Adler and Bardeen7 showed that in the PCAC limit, the simple triangle diagram is

only one which contributes to any finite order in renormalizable perturbation tlE theory . Thus within the PCAC approximation (for which there is now considerable evidence from many sources) TI0 yy becomes a unique process, since the rate can be calculated in lowest order perturbation theory despite the higher order strong interaction corrections which would ordinarily make nonsense of a lowest order calculation.

When Adler's result is used in the quark model without color the result is

o.8l eV • (8)

30 The experimental average 8 is o r(:rr 7.8 0.9 eV . (9) .... rr) experiment ± Including three colors, the prediction is increased by three in the amplitude or nine in the rate, to

7.3 eV . (10)

Of all the motivations discussed here, this is the only one which tells us that the number of colors must be three. For instance, the statistics problem could be resolved by introducing any color SU(n) with n The � 3. problems of scaling and the unobservability of quarks which are discussed below only require n ;:> 2 . I want to emphasize that it is the absence of strong interaction radiative corrections established to any finite order in perturbation theory which distinguishes this result from the multitude of other quark and parton model results and which makes it especially worthy of our attention. Deep-Inelastic Scattering c. This and the following subsection on the unobservability of quarks involve motivations for introducing color which are less direct than the

0 preceding considerations of the statistics problem and .... They are, J( yy. however, important considerations to theorists trying to understand the quark- parton puzzle and they are also essential for appreciating why color may be an especially elegant basis for quark dynamics. The quark-parton model provides a simple, intuitive "explanation" of the scaling observed in deep-inelastic scattering of electrons, muons, and neutrinos from nucleon targets. But as an "explanation" tl:e model raises questions which are more profound than the ones it answers: how can the constituents of hadrons, strongly interacting entities, behave as if they are free of strong binding forces? It turns out that this question has a simple answer in a class of field theoretical models, called "asymptotically free" theories, 9 in which the strength

3i of the interaction depends on the relevant scale of distance. When the distance scale is big (we consider that �1 is "big") the interaction is strong but f'm. for the short (i.e., lightlike) distances which characterize deep inelastic scattering the interaction becomes weaker so that the parton model is approximately valid . It was first shown10 that theories with a color symmetry are asymp totically free and, later, that they are the only such theories .11 More precisely it was shown that nonabelian gauge theories containing massless vector meson "gluons" are the only asymptotically free theories . The nonabelian symmetry of this theory cannot be associated with ordinary SU(3) which has no long range massless vector mesons associated with it . Thus we must assume a new, hidden symmetry and in view of the preceding subsections color SU(3) is a natural choice. In the version of the color hypothesis with integral quark charges, the color symmetry is broken by the term J(l,S) in the electromagnetic current, while with fractional charges the symmetry is exact . Presumably this means that there are electromagnetic corrections to asymptotic freedom in the integral charge model . However, these corrections could probably not be disentangled from high order electromagnetic effects which will produce scaling violations regardless of the underlying strong interaction theory. Therefore I doubt that asymptotic freedom alone provides a motivation for preferring the color model with fractional quark charges . D. Unobservability of Quarks The failure to observe the quark constituents is of course the central paradox. It is this paradox which suggests the analogy with the unobserved aether of the last century . Like the aether, quarks may simply mark the place where we have lost the trail . However, it is also possible that the quark idea is useful and valid even though single 300 MeV quarks do not exist. Two possibilities which have been proposed are (1) that free quarks may exist though with masses much greater than 1 GeV or (2) that quarks are permanently confined

32 within hadrons . The color hypothesis provides a natural dynamical setting for these speculations .

The possibility that quarks may be extremely massive as a consequence of color dynamics was suggested by Nambu very soon after the first formulations of 1 the color hypothesis. 2 Nambu speculated that if the color interaction is very strong and is mediated by an octet of colored gluons (gauge vector mesons) then color nonsinglet states and, in particular, quarks would acquire very large masses, say >> 1 GeV, as a result of the large self-energy in the colored field. Ordinary hadrons, being color singlets, would presumably not acquire large masses by this mechanism . This speculat ion has been verified in the semi 1 classical approximation to a quantum field theory 3 but the effect of quantum fluctuations has not been explored .

Nambu's speculation is easily illustrated if we imagine an SU (2 ) world, in which the quarks are isodoublets of ordinary SU(2 ) and color SU(2 ).

We suppose quarks interact pair-wise according to a color vector potential and that they have a very large mass M >> 1 GeV. Then the mass of a composite

system of n quarks is given by the sum of the quark masses plus the energy of

the pair-wise color interactions: n E nM t. ·t. (11) + w l. J i, j=l r.ih

where is a very large energy to be specified in a moment . Now we may write w n n n -+ 2 t t . ·t. t . \ i ·tj L l. J l. i, j=l i, j=l i=l i-fj L L {� •,) nt2 1 1 T(T + 1) - l)n • + (12) 2 2

33 where T is the total color isospin of the composite system . Substituting (12) into (11) gives

E nm + T(T + l)W (13) where m M 3 (14) w 4 3 We choose M and W to be very large, M, W l GeV, but m M - W to >> 4 be of the order of a few hundred MeV. Then for color isosinglets (i.e., "ordinary hadrons") E is of the order �1 GeV, but q_uarks and other color nonsinglet states are much heavier . The spectroscopy of the ordinary hadrons is as if q_uarks had the light effective mass m even though a single q_uark has the much larger mass M. Of course this is meant only as a simple illustration of Nambu's idea. The actual dynamics would be much more complicated . When Nambu's speculation is implemented in a semiclassical field theory13 the colored vector gluons are given a very large mass like the other colored states . For technical reasons this probably means that Nambu's mechanism is 14 incompatible with asymptotic freedom . If we accept this explanation of the absence of free, light q_uarks, we must still face the problem of understanding the deep inelastic scaling phenomena. The dynamics of asymptotic freedom also lends itself very naturally to a 15 speculation ab out the unobservability of single q_uarks. Just as the strength of the interaction decreases at small distances, it increases at large distances. Formally at least it seems that the energy req_uired to separate a single q_uark from its parent hadron becomes singular in the large distance limit. Thus q_u arks (and also the massless color vector gluons) might be permanently confined to the interior of hadronic bound states . Schwinger showed several years ago that the fermions in Q.E.D. in one space dimension are 16 confined in this sense, but it is not known whether the phenomenon actually occurs in the three dimensional gauge theories which are of interest here. If

34 this is the way the worl6 constructed, then quark and gluon fields will be is useful and experimentally meaningful concepts (e.g., in deep inelastic phenomeIB ) even though quarks and gluons do not exist as single particle states .

IV. EXPERIMENTAL TESTS

In this section I will present the short list of experimental tests of the color hypothesis which are known to me . All are tests which apply below the threshold for the production of color nonsinglet states. Most do not distin guish between the fractional and integral charge versions of the color hypothesis. However, in subsections D and E discuss experimental tests which I can distinguish between the models with fractional and integral quark charges .

It is important to keep in mind that these predictions depend not only on the color hypothesis but also on certain hypotheses about quark-hadron dynamics. The latter hypotheses are not identical for all the predictions + - considered. For instance, the predictions for o( e e y* �hadrons ) in Section A and for o(yy* �hadrons ) in E depend on the parton model/light-cone algebra hypothesis . The prediction for o pp � + - + ··· discussed in ( µ µ ) Section C depends on an extension of parton model ideas which is more spec ulative than the application of parton ideas in Sections A and E (for instance, this prediction cannot be obtained from the light -cone algebra ) . The predic tions of Section D, which test the Adler anomaly, do not require the validity of the parton model at all: they are valid to any finite order in any renor malizable field theory. However, they do require extended application of the

PCAC hypothesis.

Therefore if one of these predictions is verified, it tends to confirm both the color hypothesis and the additional dynamical assumptions which were introduced. If a prediction fails, the interpretation of the failure will depend on how it has failed and perhaps also on the out come of other related tests. For example, if the scaling law discussed in Section C for

o pp � + - ••• is not verified, it will be evidence against the application ( µ µ + ) of the parton model to this process but it will not be evidence against the

35 color hypothesis. But if the scaling law is verified and the magnitude of the cross section agrees with the prediction of the colorless quark parton model, then the result should be counted against the validity of the color hypthesis.

A. Electron-Positron Annihilation 1 According to the hypothesis of scale invariance at short distances, 7 the ratio R(s)

constant, R(s) � R, for sufficiently large values of s In the quark-parton 2 2 2 12\ (1 '1 \ 2_ . model without color, the constant is R = in \3) + \3 ) + \ 3) 3 , 1 8 the model with color SU(3 ), it is R = • 2. The experimental data � 3 shows that from s 6 Geif to s Geif R(s) is indeed constant within ';;: 12 the �1CY{o systematic errors: this evidence for scaling is at least as good as the first evidence for scaling in deep-inelastic electron scattering, which covered a similar range in Q2•

Within this range of s the average value of R is 2.45. Considering the point-to-point systematic errors of �1CY{o plus the lCY{o uncertainty in the overall normalization (which is not included in the error estimates usually displayed with the data), the experimental result is in reasonable agreement with the prediction of the color su(3) hypothesis. Furthermore asymptotic freedom predicts that R(s) should approach its asymptotic value slowly 1 (corrections are like �1/tn s) and from above; 9 the l/tn s corrections are 20 expected to be. of the order of �1CY{o to �5% in this range of s The emergence of new phenomena for s Ge if prevents us from :;:. 12 testing the asymptotic freedom prediction of a l/tn s approach to the asymptotic value of R. However, the new phenomena do offer a second possible testing ground for both the color hypothesis and asymptotic freedom. If we can describe the new hadronic spectroscopy in terms of new quarks and if the contribution of the new hadrons to the cross section can be disentangled from new nonhadronic degrees of freedom which may also be present, then we can again check whether the new quarks come in three colors. And if we are given a long enough respite before more new thresholds in s occur, we will also be able to

36 determine whether the approach to scaling is logarithmic .

The evidence for a new antibaryon of charge -2 presented yesterday by W. Y. Lee suggests that the new quark associated with the threshold in R has charge 2/3, as we expect in the G.I.M. hypothesis. Including the factor 3 for color, we expect asymptotically an increase in of R This is consistent with the data, provided we take into account the additional new contributions of the heavy lepton discussed in last weeks sessions. Although more conjectural, it is also possible that the dynamics of the new hadrons may provide insight into color and asymptotic freedom. According 21 to the charmonium model, the large mass scale of the new hadrons means that their decays are characterized by short distances and small effective coupling constants . This idea implies a variety of qualitative, semiquantitative and perhaps (if we are optimistic enough ) quantitative tests . For instance, of the three predicted p-wave excitations of �(3.1), the scalar and tensor would decay into hadrons via two color vector gluons but the pseudovector state would have to decay via four vector gluons (Yang 's theorem) and would consequently 22 have a much smaller hadronic width than its two :i;artners. This possibility will be discussed in some detail and compared with the available data by Jackson in his talk in this session. B. Violations of Scaling in Deep-Inelastic Scattering Caref'ul measurement of the deep-inelastic lepton-nucleon structure functions can in principal provide a test of asymptotic freedom and therefore, 2 indirectly, of the color hypothesis. For Q and large enough so that v perturbation theory in the effective coupling constant is valid, the theory predicts logarithmic violations of Bjerken scaling . The most rigorous predictions are for the moments of the structure functions: for instance, for the 2 proton-neutron difference in electroproduction the result is 3

37 ,. 1 / n

i n-2 I=l 2 I x vw (x, Q ) n "\ dx 2 cn 1 + (15) [ O )0 � :: ) (in\�] where

2 4 t..n n(n + 1) + (16)

2 Here x = Q / M v is the scaling variable, e is an undetermined constant, 2 p n 2 µ is a constant+ found experimentally to �>be �0(1} Ge,j?- ), and A is a constant which is determined by the color gauge group . For color su(3), A = 2/27. Data from SLAC are compatible with the predicted logarithmic violations and

with the value of A predicted by color SU(3), though the uncertainties are 24 too great and the kinematical domain too restricted to be conclusive .

The equations for the moments, such as (15 ), can be inverted to give a qualitative--and perhaps even a quantitative--picture of how the structure 2 functions change with increasing Q • Using positivity and the dimension of the stress tensor it is known that for interacting field theories �=0 and 2 25 This implies that in a plot of versus x, as Q in- t..n+l t..n • > creases the area under vw remains constant while vW becomes more sharply 2 2 peaked in the small x region. The most recent data from the FNAL muon 4 experiment does show this pattern of scaling violations .

With an additional dynamical assumption the equations for the moments 26 can be inverted and solved for the structure functions. The structure

functions at all Q2 are then predicted from their values at some initial 2 2 2 Q = Q and by their value at one other (x, Q ) point to determine the 0 ( 2 parameter µ ) . The results of such analyses are again compatible with the SLAC data and with the recent FNAL data. However, it has been argued that the

data is also consistent with a phenomenological treatment of nonasymptotically

free theories with anomalous dimensions .24 Enormously larger values of Q2

would be necessary to distinguish between these two possibilities .

38 Production of Massive Muon Ri.irs in Hadronic Collisions c. 7 Drell and Yan have argued2 that the :i;a.rton model may be applied to the process

- proton + proton + -+ µ µ + provided that the variables

s (p )2 proton Pproton 1 + 2

satisfy the constraint

- s Gel (17) » 1 with Q2/s finite . In this kinematical region they argue that the impulse approximation is valid and that the dominant process is the annihilation of a

parton from one proton with an antiparton from the other proton. In the

scaling limit in a :i;a.rton model without color their expression for the inclusive muon :i;a.ir cross section is

4rra2 dx [ l [ 7 0 0

where is the charge of :i;a.rton i and are the

probabilities to find the i'th parton or antiparton with a fraction xj of the longitudinal momentum of proton j (in a reference frame in which the protons are moving very rapidly, with momentum P >> "Vs). The F (x ) may be 2i extracted from electron, muon, and neutrino deep-inelastic scattering data .

The first experimental test of (18) is to check the scaling law: 4 should be proportional to Q- times a function of the scaling variable

If the scaling law is verified, we may assume that the parton-antiparton

annihilation mechanism is dominant and we may proceed to study (18 ) quant ita- tively. It is at this point that we can test the color hypothe sis . If the

quark-partons come in n colors, then an additional factor of l/n appears on

39 the right side of (18) to account for the diminished probability that a r:arton of a given color in one nucleon will find its antir:arton color-mate in the other 28 nucleon . Reports from an FNAL neutron scattering experiment appear to be consistent with (18) . In this session Quigg will discuss the most recent available data concerning the scaling and magnitude of the cross section.

The dilepton production cross section has an important advantage in the - search for a high-energy color threshold . In e+e annihilation we cross the + - color threshold when the mass of the e e r:air is larger than the threshold + - energy. But in pp � we cross the threshold when the invariant mass µ µ x M,c of the hadronic state recoiling against the dilepton is larger than the threshold energy . This means that we can utilize a very large fraction of the available energy in hadron-hadron collisions to search for a color threshold .

D. Further Tests of the Chiral Anomaly

As discussed in the previous section the low-energy theorem for n° � provided one of the princir:al motivations for the introduction of rr color and was the only motivation requiring that the color group be SU(3). This is a considerable theoretical weight to rest on one experimental number.

Furthermore several authors have suggested that the PCAC extrapolation from zero pion mass to the mass shell may account for the missing factor of three in the amplitude and that there is no need to introduce color to explain the decay rate. 29 It is therefore important to find other tests of the physics which underlies the low-energy theorem and of the strong PCAC assumption that the corrections due to the mass shell extrapolation are small.

Fortunately there is a large family of related anomalies, 30 corresponding to triangles, boxes and pentagons, which gives rise to low-energy theorems which are also free of strong interaction radiative corrections. For

instance, the r:art of the decay amplitude � which proceeds Kt4 (K nntv ) through the weak vector current is controlled by such a low-energy theorem . A comr:arison with experiment indicates agreement with the anomaly, strong PCAC

40 (which here includes an extrapolation to zero K mass), and color su(3 ), 31 though the uncertainties are quite large . I have recently considered the low-

energy theorems for and and find32 that the experimental ri � rr result s are again consistent with the anomaly, strong PCAC and color SU(3 )

provided that the has a small, negative singlet-octet mixing angle as in the T) quadratic ma ss formula . These result s are of additional interest because they

may enable us to distinguish between the two versions of the color hypothesis

even though we are below the threshold for the production of color .

Consider first the reason that rt0 � does not distinguish the two yy kinds of color . In the model with fractional charges, the amplitude is 2 (rt0 (J ) where denotes the vacuum and J B was defined in 1 (8, 1 ) lo) lo) ( ,l) 2 Section II . With integral charges, the amplitude is ( ) (rt0l J(8,l) - J(l,8) l o) and the difference between the two models is given by

- • Now is a member of (rt0 1 - J (8,l)J (l,8) J (l, 8 l(B,l) + i(l, 8) lo) rt0 (8, 1) (the 8 of ordinary SU(3 ) and the 1 of color SU(3 )) but (8, ) (1,8) (8,8) has no projection into the of color SU( ) and 1 x 1 3

(1 8) (1,8) (1,1) (1,8) ••• has no projection into the 8 of ordinary , x + +

su(3 ). Therefore the difference term vanishes and the amplitude is the same

in both models .

However, now suppose that we consider the decay of an SU(3 ) YY singlet pseudoscalar meson . Then, as was first ob served by Okubo, 33 the

amplitude in the integral charge model would be

(19)

which is exactly twice the amplitude obtained the fractional charge model . in (This is because (8,1) )( (8,1) and (1,8) have equal proj ections (1, 8 ) X int o (1,1).)

41 ° 2 Even for small mixing angles (e.g., -11 ), the analysis3 of e =

� and � indicates that the singlet component of is responsible � yy � nnY � for about half of the observed rate. That is, for a pure octet we have

0 � = 160 eV but with a mixing angle of -11 the result is nearly r(�8 yy) e 34 300 eV. The most recent experimental value is 324 46 eV. � Since the singlet component makes a sub stant ial contribution to

� we may hope to use the decays to distinguish between the two � yy, � versions of color . In the analysis it is crucial to consider simultaneously

the low-energy theorems for � � and � � in order to resolve the yy nnY uncertainties due to our ignorance of the SU(3 ) singlet PCAC constant (the

analogue of F MeV) and the mixing angle .32 The results of this analysis n 95 are compatible with strong PCAC and color su(3 ). The fractional charge verskn

of the color hypothesis is preferred, though the differences between the

predictions of the two color models are too small and the uncertainties are too

great for this to be decisive . It is important to note that this is a non-

trivial �est of the anomaly because it involves not only the familiar triangle

diagram but also the box diagram .

In order to clearly distinguish between the two color models, it is

necessary to consider a further step into the uncertainties of large mass PCAC

extrapolations and to consider the low-energy theorems for � ' and YY

� ' � Since the is taken to be dominantly in an SU(3 ) singlet, it nny. �' is most sensitive to the differences between the two color models . Making use

of the constraints obtained in the study of and - we find � - yy � nny, that for any value of the mixing angle in the interval 0 17°, the < -8 -$ ratio of decays is �·

for fractional charges and

42 8 0 1. > for integral charges . The experimental situation is confused with experimental 5 values report�d3 between 9.1 and 17 3 (the confusion is due to t 2.2 t disagreement on the branching ratio for D' -+ rrny). The fractional charge model is much nearer the range of experimental values is compatible with the anl. lowest reported value .

It is necessary to emphasize the uncertainties this analysis: in D' pole dominance may or may not be valid in these amplitudes and PCAC for the 36 SU(3 ) singlet axial current involves theoretical problems which are not understood . Since theoretical ignorance in these areas is nearly perfect, my attitude is by necessity purely phenomenological . To go farther we need exper imental measurements of the rates r(D' -+ yy ) and r(D ' -+ rr+n-y), to see if these rates and their ratio are compatible with the low-energy theorems . These rates and the total width of the �· are presently not known. The present bounds are based primarily on strong interaction bubble chamber experiments . Primakoff photoproduction at and SPS energies and FNAL production by the two photon process in e ± storage rings are also alternatives which should be considered.

There are also low-energy theorems for the yrrrrrr and yynrrrr 37, 3S amplitudes which test the theory of chiral anomalies and color SU(3 ),

0 though like -+ they do not distinguish between the two color models . rr YY These amplitudes could be studied in e + e - colliding beams . In order to minimize the extrapolation to the low-energy point (where all momentum invariants vanish, p p = 0) it is important to make the measurement as near i · j the threshold, s 2 as possible . If for instance a measurement could be = 9mrr ' made at s = 16m , then the extrapolation to s = 0 would be comparable to rr2 the extrapolations re�uired in the and � decays discussed ab ove . K

43 The amplitude can probably be studied more favorably by using a ynnn pion beam to produce a dipion the Coulomb field of a heavy nucleus, that is, in 39 + - --...). the Primakoff effect. This has a clear advantage over e e /--).rrrrrr since in the Primakoff process the photon mass is very nearly zero. To minimize the other momentum invariants and the effect of the rho meson tail, the measurement 2 should be performed as close to the dipion threshold, s = 4m , as possible . 1(1( 1( In order to achieve a small photon mass and a pronounced Primakoff peak, the experiment would best be performed at FNAL or SPS energies. An alternative 4 method which has been studied recently 0 is pion-electron scattering, again at very high energies. This method has the advantage that it lacks a strong interaction background . The low-energy theorem for the two photon decay of the scalar resonance, 41 also provides a test of the color hypothesis. With the usual assignment of the E to an su(3) singlet, this decay is also sensitive to the difference between fractional and integral charge color models . The result is

2 0.2 R keV (20)

where R = 2 for the color model with fractional charges and R = 4 with integral charges. This low-energy theorem is not free of radiative corrections in perturbation theory. Rather its validity depends on the same assumptions ab out short distance behavior which are implicit in the parton model and the light cone algebra. The measurement of could be performed using the E � yy two photon process in e ± storage rings or perhaps by Primakoff photoproduction. Preliminary indications based on a very few events are consistent with the rate predicted by the low-energy theorem with color, though the 42 uncertainties are too great to distinguish color models. Furthermore, equation (20) is based on a narrow-width pole dominance extrapolation from the low-energy point; for quantitative comparisons it is important to do a more careful treatment of the extrapolation which takes account of the large width

of the E.

44 E. Deep Inelastic Scattering from a Photon Target Electron-positron storage rings can be used to study photon-photon scattering by selecting events in which the electron and positron do not 4 annihilate. 3 This cross section is smaller than the annihilation cross secticn 2 by a factor a but when the photons are near the mass shell the cross section 2 is enhanced by a factor -(ln E/m e ) • This together with the strong forward peaking of the events means that it should be possible to measure yy

- hadrons ), probably in the next generation of storage ring experiments. a( n In the parton model (or, equivalently, using the light cone algebra ) the cross section for two highly virtual photons to scatter into hadrons is given by the easily computed diagram for the two photons to annihilate and 4 create a quark-antiquark pair. 4 This cross section contains a factor e 4 L.i i where the are the quark charges, allowing us in principle to distinguish between the following possibilities:

! 2 No Color + + 1-1)4 \3 9

6 Color-Fractional Charges 4 9 e i 2-:i 18 Color-Integral Charges 9 (below color threshold )

36 Color-Integral Charges l 9 (above color threshold )

For the same group theoretical reasons discussed in connection with the � and �· decays in Section D, the scattering of two highly virtual photons allows us to distinguish between the two versions of the color hypothesis at energies below the color threshold . Unfortunately this is of purely theoretical interest, since when both photons are very virtual the enhancement [ , 2 tn(E/m ) are replaced by suppression factors due to the photon factors e J

45 propagators and the cross section is far too small to observe . 45 However, R. P. Worden has shown in a beautiful analysis that it may 4 be possible to measure e in processes in which only one photon is I:i i virtual while the second is near its mass shell . Worden obtains this result by using analyticity together with the light-cone parton model results of Ref. 44.

The transverse structure function, WT' is controlled by the scat tering of the virtual photon from the hadronic (vector dominated ) components of the on-mass-shell photon; therefore W is not sensitive to the presence of T color . But the longitudinal structure function W L and a second structure function (which appears because the photons tend to be polarized ) are w3 dominated by parton-antiparton creation and are therefore proportional to 4 e . . The structure functions may be separated experimentally by char- Zi 1 acteristic dependences on the photon polarizations, and in the case of by w3, a characteristic angular dependence. Since only one photon is virtual, the possibility of observing this process is greatly improved . It would be interesting to explore the fea- sibility of measuring this process either at present high-energy storage rings or in the next generation of storage rings with beam energies of � l5 GeV.

A final remark concerning the prediction lB/9 for the case of color- integral charges (below color threshold ): this assumes that the color singlet projection of J2 contributes fully . This is a nontrivial assumption about (l, B) the dynamics which requires investigation . The analogous assumption in the analysis of and decays (Section D) is justified by the fact that the � � · chiral anomaly is known to probe the ultra-violet behavior of the underlying theory. For the process ;N -+ ;X I have shown that in a restricted ("quasi- forward ") part of the scaling region, parton dynamics ensures that the singlet projection of does indeed contribute fully (this work will soon be submitted for publication .). A similar analysis has yet to be done for ee - eeX.

46 V. CONCLUDING REMARKS Though it has not been established that color is an absolutely nee- essary element of the quark model, it is a very natural element . If the color hypothesis is shown to be inconsistent with experiment, it will reflect on the validity of the quark model itself unless alternative explanations of the problems discussed in Section III can be obtained . Present experimental knowledge is consistent with the color hypothesis but more experimental information should be forthcoming as discussed in Section IV. Some of the experiments discussed in Section IV--e.g., those involving storage rings or deep-inelastic scattering--will soon be performed since they are of large ongoing exper- :i;art imental programs . Other experiments, equally worthy of study, have not yet, to my knowledge, been placed on an experimental program . In this category I have in mind especially the Primakoff experiments discussed in Section IV which test

the physics underlying the low energy theorem for 0 :Jl .... "/"/ .

FOOTNOTES AND REFERENCES

* This work was supported by the U. S. Energy Research and Development Administration.

1. O. W. Greenberg, Phys. Rev. Lett . � 598 (1964). W. A. Bardeen, H. Fritzsch, and M. Gell-Mann, Scale and Conformal Symmetry 2. in Hadron Physics, ed. R. Gatto (Wiley, New York, 1913 ). 3. M. Y. Han and Y. Nambu, Phys. Rev. Bl39, 1006 (1965 ); Y. Miyamoto, Prag . Theo. Phys. Sup . Extra No., 18[ (1965 ); A. Tavhelidze, Proceedings of the

Seminar on High Energy Physics and Elementary Particles, I.A.E.A., Vienna,

4. Y. Watanabe et al ., Lawrence Berkeley Laboratory Report LBL-3870 (1915 ) ; C. Chang et al., Lawrence Berkeley Laboratory Report LBL-3886 (1975 ). 5. For a complete review with references to the original literature see, S. L. Adler, Lectures on Elementary Particles and Quantum Field Theory, ed . s. Deser et al . (M.I.T. Press, Cambridge, Mass., 1911 ).

47 6. S. L. Adler, Phys . Rev. 177, 2426 (1969); S. Bell and R. Jackiw, Nuovo J. Cimento 6oA, 47 (1969).

7. S. L. Adler and W. A. Bardeen, Phys. Rev. 182, 1517 (1969).

8. Particle Data Group, Phys . Lett . � 1974 . 9. For a recent review, see H. D. Politzer, Physics Reports 14, 129 (1974).

10. G. 't Hoo�, 1972 (unpublished ); D. Gross and F. Wilczek, Phys . Rev. Lett .

§£, 1343 (1973 ); H. D. Politzer, Phys . Rev. Lett . 26, 1346 (1973 ). 11 . S. Coleman and D. Gross, Phys. Rev. Lett . I!:,. 851 (1973 ). 12 . Y. Nambu, Preludes in Theoretical Physics, ed . A. de Shalit et al . (North

Holland, Amsterdam, 1966 ).

13 . W. Bardeen, M. Chanowitz, S. Drell, M. Weinstein, and T.-M. Yan, Phys . Rev.

� 1094 (1975 ). 14. See reference 9, p. 148, and references therein .

15 . See, for instance, A. Casher, J . Kogut, and L. Susskind, Phys. Rev. Lett .

_g,, 792 (1973). 16. J . Schwinger, Phys . Rev. � 2425 (1962 ).

17. K. Wilson, Phys . Rev. 179, 1499 (1969).

18. J . E. Augustin et al., Phys . Rev. Lett . 34, 764 (1975 ).

19. T. Appelq_uist and H. Georgi, Phys . Rev. D8, 4000 (1973 ); A. Zee, Phys . Rev. � 4038 (1973). 2 20. The leading corrections are ��n � , where the scale µ � 1 Ge.f'- is µ suggested by studies of the electroproduction data (see Section IV.B).

21. T. Appelq_uist and H. D. Politzer, Phys . Rev. Lett . 34, 43 (1975 ).

22 . This remark is due to Ken Lane .

23. D. Gross and F. Wilczek, Phys . Rev. D8, 3633 (1973 ); H. Georgi and H.

Politzer, Phys . Rev. D9, 416 (1974).

24. See, for instance, the analysis of W.-K. Tung, Un iv. of Chicago preprint

EFI 75/3 6, July, 1975, and references therein ..

25. 0. Nachtman, Nucl . Phys . B63, 237 (1973).

48 26. G. Parisi, Phys. Lett . 43B, 107 (1973 ) and 50B, 367 (1974). See also Ref. 9, p. 167.

27. S. Drell and T.-M. Yan, Phys. Rev. Lett. � 316 (1970). 28. W. Y. Lee, reported to the 1975 SLAC Summer Institute on Particle Physics, to be published. 29. S. Drell, Phys . Rev. D7, 2190 (1973 ); R. Delbourgo and M. Scadron, Univ. of Arizona preprint (July, 1975). 30. W. Bardeen, Phys . Rev. � 1848 (1969); R. Brown, C.-c. Shih, and B.-L. Young, Phys. Rev. 186, 1491 (1969). 31. Calculation by J. Wess and B. Zumino, reported in L.-M. Chounet, J.-M. Gaillard, and Gaillard, Phys . Reports 201 (1972 ). M. K. ,'.:£, 32 . M. Chanowitz, Radiative Decays of and as Probes of Quark Charges, � � · lawrence Berkeley laboratory report LBL-4200 (July, 1975 ). 33 . S. Okubo, Symmetries and Quark Models, R. Chand (Gordon and Breach, ed. New York, 1970 ). See also H. Suura, T. Walsh, and B.-L. Young, Lettere al

Nuovo Cimento _'.:, 505 (1972). 34. A. Browman et al., Phys. Rev. Lett . 32, 1067 (1974). 35 . See Aguilar et al., Phys. Rev. R§., 29 (1973 ); A. Rittenberg, lawrence Radiation laboratory report UCRL-18863, Thesis (1969), and Ref. 8. 36. See S. Weinberg, Proceedings of the XVII International Conference on High Energy Physics, ed . J. Smith (Science Research Council, London, 1974). 37. M. V. Terent 'ev, JETP Letters 14, 140 (1971 ) and Thysics Letters 38B, 419 (1972 ). 38. S. Adler, B. Lee, S. Treiman, and A. Zee, Phys . Rev. D4, 3897 (1971 ). 39. This has been studied by A. Zee, Phys . Rev. D6, 900 (1972 ). See also Ref. 37. 40 . R. Crewther and s. Rudaz, unpublished . 41 . R. Crewther, Phy s. Rev. Lett . 28, 1421 (1972 ); M. Chanowitz and J. Ellis,

Physics Letters � 397 (1972 ) and Phys. Rev. !21- 2490 (1973 ).

49 42 . S. Orito, M. Ferrer, L. Paoluzi, and R. Santonico, Physics Letters 48B, 380 (l974).

43 . For a review with references see H. Terazawa, Rev. Mod . Phys. 45, 6l5 (l973 ).

44. D. Gross and S. Treiman, Phys . Rev. D4, 2l05 (l97l ); T. Walsh and P. Zerwas, Nucl. Phys. B4l, 55l (l972). For anore complete list, see Ref . 45 .

45 . R. P. Worden, Phys. Lett . � 57 (l974).

50 UNBOUND COLOR,

PREFACED BY REMARKS ON BARYON SPECTRO SCOPY

Greenberg* 0. W. Center for Theoretical Physics Department of Phys ics and Ast ronomy UNIVERSITY OF MARYLAND College Park, Maryland 20742 U.S.A.

I survey theoretical and experimental issues related to the possibility that color is unbound; i.e. that quarks, gluons and other particles carrying color can exis t as isolated obj ec ts. I point out that it is surprisingly difficult to dis tinguish models with unbound color from those in wh ich color is permanently confined. None-the-less , the present situation seems discouraging for unbound color because there is no unambiguous support for it and because the cruc ial prediction of fo rmation of a co lored gluon in e+e- collisions has been ruled out wherever sufficient data exists.

I preface the above survey by remarks on the symmetric quark mod el for baryon spectroscopy.

Je fais le releve des questions theoriques et experimentales concernant la possib ilite que la couleur est non li€e ; c'est-3-dire que les quarks , gluons et autres par ticules possedant de la couleur puissent exister comme obj ects isoles . Je montre qu' il est etonnamment di fficile de distinguer les mo deles avec la couleur non liee des modeles dans lesquels la couleur est confinee de f a�on perrnanente. Neanmoins , la situa tion actuelle apparait decourageante pour la couleur non liee parce qu' il n'exis te pas de corroboration inarnbigue pour cette hypothese et parce que sa prediction cruciale de la formation d'un gluon colore dans les collisions e+ e- es t en coHtradiction avec l'experience partout oil il exi ste suffisamrnent de donnees.

Dans l'introduction, je fais des remarques au suj et du mo dele symmetrique des quarks et de la spectroscopie des baryons.

* Supported in part by the National Science Foundation.

51 I am going to spend most of my time talking about models with unbound

(l) color ; I'll try to give as specific comments as I can about the present

status of such models . I will preface this discussion by some comments on

baryon spectroscopy using the symmetric quark model.

1. BARYON SPECTROSCOPY

I start by reminding you of the charges of the nonets of quarks in the

( Z) (3) paraquark or identical three-triplet models :

fl a n 1 v� p 2 2 2 p 3 3 3

1 1 1 n - Q -3 -3 3

1 1 1 A - - 3 -3 3

q - (3,3*) under SU(3) x SU(3) (f stands for flavor, c stands for color) f c

- y (8,1)

Identical three-triplet = paraquark model

and in the Han-Nambu model (4):

color- averaged r fla p n charge v� 1 2 p 0 1 1 3

1 Q -1 0 0 n -3

1 A -1 0 0 3

q (3, 3*) under SU(3) SU(3) - f x c

(8,1) + (1,8) y -

Han-Nambu model

52 The illlportant thing to note about these two models is that in the identical (5) three-triplet model the photon is a pure color singlet (and flavor octet) , while in the Han-Nambu model the photon is an equal superposition of color- singlet (flavor-octet) and color-octet (flavor-singlet) terms . (6) Both of these models are equivalent for baryon spectroscopy and for

color singlet states in most cases. As far as the static properties (masses, magnetic moments, etc. ) of the known baryons , which are all color singlets , are

concerned,only the color-averaged quark charges, which are equal to the

fractional charges of the single-triplet Gell-Mann-Zweig model, are measurable.

Including the ordinary spin, the quark can be considered to transform as the

fundamental 18 representation of SU(l8) , and has the following chain of decompositions under the indicated reductions:

q - 18 + (6,3*) + (3,3*,2), under

SU(l8) + SU (6) x SU(3) + SU (3) x SU(3) x SU( ) f c f c 2 s

Using the generalized Pauli principle, the color singlet qqq states must be

( ) x ( ) x SU(l8) space SU (3) § c

For the ground state baryons , the space wave function, s� is totally symme tric,

, and thus the SU(6) wave function must also be symmetric, 56. [I]] CIJJ = ( Gilrsey and Radicati ?) gave the decomposition

+ CIJJ (g:i x x �6 (8, 2) + (10 , 4) •

under SU(6) + SU (3) SU ( )s which gives the reduction of the 56 of SU (6) into f f x 2 the lowlying spin-1/ SU(3) octet and spin-3/2 SU(3) decouplet . known 2 In general, the reduction of the space and flavor SU(6) wave function must be totally symmetric under permutations of the quantum numb ers associated with

53 the three quarks. Thus three possibilities are available:

space x SU(6)f OIJ

x 56 ITIJ QI] EP x EP 70

x 20 § Up to now all presently§ known and assigned baryon resonances fit into the SU(6) f 56 or 70; however, for two reasons the possible existence of baryons in the 20 is not ruled out. First, it is hard for baryons in the 20 to be excited from

tne baryons in the 56, due to the necessity of a double SU(6)f and space synnnetry rearrangement. Secondly, the states in the 20 are difficult to find experimentally, because they tend to have a low angular momentum because they are associated with totally antisynnnetric space wave ftmctions. Resnikoff and (7) I began the systematic study of these SU(6)f baryon multiplets using SU (6) tensor analysis. The most recent work of this kind has been done by Dalitz and S Horgan( ) and by Jones, Dalitz and Horgan.( g) One can label the supermultiplets by

where the SU(6) representation is labeled by its multiplicity, L is the total

orbital angular momentum, P is the parity, and N is the number of harmonic oscillator quanta. Jones, Dalitz and Horgan made a detailed fit to the representations (56,0+ (70,1- ) , (56,2+ ) and (70,2+ ) , including SU(3) )0, 1 2 2 breaking, and spin-orbit and tensor noncentral potentials, with very good results. Other representations in which baryon resonances have been assigned i. nclude (56,0+ ) , (70,0+ ) , (70,3- ) , (56, + ) , and (56,1- ) • The harmonic 2 2 3 4 4 3 oscillator model is the simplest to calculate. Horgan has introduced a group theory analysis of the two traceless orbital modes which occur in the three-body

54 system (without center of mass motion) using a "dynamical" SU(6) group . Cutkosky and Hendrick(lO) have used a model with a linear string potential,

together with a Coulomb potential, to do a less detailed analysis of the multi

plets. In their analysis they take the mass of the highest-spin 6 resonance as the mass of the supermultiplet, and do not consider the splitting of the states within each supermultiplet. Their main qualitative result is that radially excit ed states are lowered in energy relative to orbitally excited states with the same (ll) (l2) N. De Rujula, Georgi and Glashow, and Gunion and Willey have studied baryon spectroscopy using models based on QCD. Chodos, (l3) et al. , Rebbi, (l4) and R. Jaffe(lS) have studied baryon spectra in the MIT bag model; Jaffe, in particular, has emphasized the possibility of crypto-exotic states; that is states whose quark composition is exotic, but whose SU(3)f characteristics are normal. The systematic SU(6) tensor analysis of mass operators is less economical than the QCD and MIT bag analyses; however the latter have not been applied to study the detailed splittings of higher multiplets. The rules of the SU(6) tensor analysis are as follows : assume that the masses within a given supermultiplet are split by terms linear in the SU(6)f breaking mass operators . Allow as SU(6)f-breaking terms all operators which can be constructed as one- or two-body quark operators which transform as an

SU(3)f singlet or as the I=Y=O member of an SU(3)f octet , and which transform as spin zero, one or two operators. In the case of the non-central operators, form rotationally invariant operators by contracting the S=l or 2 quark-spin operators with L = or 2 orbital operators . Various �wo-body operators can be 1 used to describe the masses of more than one supermultiplet , so 'that the entire scheme , although still phenomenological, is relatively predictive . I will sketch a little bit of the detail of this analysis . From the point of view of the symmetric quark model, one can suppress color SU(3) and simply take the quark to be in states which are totally symmetric under permutations of the SU(6) and orbital degrees of freedom of the three quarks . For the SU(6) f f analysis, it is sufficient to consider two-body operators, because they include

55 the tenns which occur as one-body operators. The most general two-quark SU(6) f state can be decomposed into symmetric and antisymmetric two-quark irreducibles of SU (6) according to: f

6 6 21 + 15 , in SU(6) . x f + rn B

The mass operators do not mix states in these two representations , because these representations are associated with orbital states of opposite parity . Thus the most general two-body quark operator must be a component of the two-body tensors which occur in

1 35 405 ' 21 x 21* + + for the symmetric two-quark SU (6) states, or in f

1 + 35 189 ' 15 x 15* + for the antisymmetric two-quark SU (6) states . For the former case the available f tensors are:

2 1,1 2 8,1 21 1,1 2 8,1 � � • ' T , or � 1 ' 35 405 405 for the central forces , as well as

21 1,5 21 8,5 T or T 405 405 for the tensor forces . The notation here is

2-body SU (6) f SU(3) , SU ( 2) T f s SU(6) f whe re the representations are labeled by their dimension . A similar analysis holds for the antisymmetric two-quark states .

The results of this analysis have been very successful in accounting for the overwhelming majority of baryon resonances : Not only are many masses

56 predicted using a relatively small number of input masses to evaluate reduced matrix elements, but, in addition, the mixing parameters of states with the same

I, Y and J within a given supermultiplet are determined, and these mixing parameters can be used to predict decays. In general, decays of resonances on the leading trajectory (with high spin) are well predicted; however daughter trajectory (with low spin) decays are poorly predicted. It is likely that relativistic corrections play a large role in the latter cases . For charmed baryons an analogous study can be made using

+ + ' [J]J ) [IIJ ( [IIJ 2 + 20 120 + s 4 under

x SU(2) + s ) This analysis has been outlined by Dalitz(lG at the Oxford baryon conference last Stlllllller and is being pursued by Dunbar.( l7) Other authors , such as S Lichtenberg, (l ) Nelson(l9) and others have also pursued the study of charmed baryons. Saturation is the name given to the fact that the low-lying hadronic states can all be accounted for in terms of three quarks for baryons or quark and anti quark for mesons. Zwanziger and I( 3) studied a variety of saturation mechanisms in various triplet models to see how saturation can come about. (20) Our conclu sion was that the original single-triplet model of Gell-Mann and Zweig does not produce saturation. The most appealing mechanism for saturation was given earlier by Nambu(4) using exchange of a colored octet of gluons to make the lowest-lying states color singlets. Note that if the quark is a 3* (or· a 3) under color, then the color-singlet states with the fewest particles are qq and qqq for mesons and baryons , respectively. Mike Chanowitz discussed the Nambu mechanism in the context of SU(2)c (for which baryons would be qq) this morning; here I remind you of the situation in the context of SU(3) � . The mass of a system of a total of N quarks and/or antiquarks is given by

57 8 M(N) NM U q + t E Ia(i) Ia (j ) a i

effective quark mass for quarks bound in hadrons, U is a large, positive

potential strength parameter, and C is the quadratic Casimir operator of SU(3) , 3) (sooet i.oes written c� ). The values of C for low multiplicity SU(3) represen-

tations are :

Representation 1 3 or 3* 6 or 6* 8 10 10 c 0 4 3 6 3 3

If_ m is small and U is positive and large , then the color singlet is the lowest q state and !:"he a d 3* which are next--e=--he-.made--Y.er:)'_mass_ive . The octet is 3 n

higher than the trip let; i.e. colored gluons which are composites of quarks

and antiquarks are more massive than quarks ; however elementary colored gluons

can have any ma ss.

. IS CONFINEMENT POPULAR? 2 WHY (a) The most obvious reason for the popularity of color confinement is

the fact that no fractionally charged quarks have been found despite intensive

searches ranging from cosmic rays in the upper atmosphere to oysters at the

bottom of the sea. (2l) In models with unconfined color, quarks can be integrally

charged, and with proper choice of baryon number assignment can be uns table even

if baryon numb er is conserved. (22) Alternatively quarks can decay via baryon (Z3l number-non-conserving interactions .

(b) Asymp totic freedom leads to a field theoretical basis for the parton model and logarithmic corrections to it. As is well-known , asymptotic freedom

holds for unbroken non-abelian gauge symmetries , such as SU(J) . c

58 theories in which a single coupling constant governs the interaction strength

of all interactions at a sufficiently large mass scale, typically of the order -1/2 19 -5 of the Planck mass G (10 GeV , which is approximately 10 grams(24) ) . The N

Planck mass also figures as the mass necessary to make the proton sufficiently

stable in some unified models in which baryon conservation is violated.

(d) There are no compelling theoretical arguments for unbound baryon

color in contrast to the compelling GIM argument for charm as necessary to

suppress strangeness-changing neutral currents . (So far , however, no mechanism

for permanent color confinement has been demons trated .)

(e) There· is no compelling experimental argument made for unbound color

in contrast to the J/¢, ¢' , x's, D's, D*'s ••. for which charm is necessary .

3. THEORETICAL PROBLEMS FOR MODELS WITH UNBOUND COLOR

(a) The menace of U(9) degeneracy : ( 25) The strong interactions must be mediated by a color-octet of gluons , rather than by a color-singlet gluon,

because mediation by a color singlet leads to a U(9) degeneracy of the hadron

spectrum. Spontaneously broken �ymmetry cannot solve this problem , because massless Goldstone bosons always remain. In addition , if the weak interactions

obey a local gauge symmetry C before· spontaneous symmetry breaking and ,,

diagonalization of the gauge boson mass matrix, then C must be a singlet under ,,

color to allow strong interactions to be mediated by a color octet of gluons .

I will sketch the argument for this conclusion:

Let

L L + + kinetic L strong where L + part of L is invariant under the local gauge symmetry G . weak kinetic W

Here L is renormalizable, but is not necessarily a gauge Lagrangian. Let strong

Q be the generators of G . Then i W

+ q � q + non-quark terms ,

59 wh ere q is the nonet of quark fields, and the � are matrices. Let

g q a terms which commute with G , L strong r X q V + W s a j j where are in the algebra of Dirac matrices, and are in the algebra of r X a j SU(3 x SU(3) matrices acting on the quark nonet. To preserve G , we need >t c W

or [y0f X. , 0.) 0, o, a J 'i where ET stands for equal times. If

1 c where A's are the relevant Gell-Mann matrices, then G is preserved. No other W solution preserves G and SU(3) . Thus before spontaneous symmetry breaking , W c the weak gauges must commute with SU(3)c

Natural conservation of parity and strangeness to order a also requires

The implication of these results is that one must introduce charm in order to avoid strangeness-changing neutral currents. One cannot pick members of the Han-Nambu quark nonet by hand to produce a GIM mechanism. For example, with

1 2 3 p 1 n Q 0 -1 0 : ) 2 2 3 3 the replacement (_(p,:n, A,c) + (p ,n ,A ,p ) has the correct set of charges to give a GIM mechanism, but leads to a conflict with the hadron spectrum. (b) Temporary asymptotic freedom: The problem is that, in 3 space 1 + time , non-abelian gauge theories are the only ones with asymptotic freedom and positive energy. Once the symmetry is spontaneously broken asymptotic freedom

likely to be lost. A possible way out of this difficulty is to make use of is

60 the fact that the renormalization group equations couple gauge and quartic

scaling effective coupling constants, so that, in principle, complete Higgs-

Kibble symme try breaking might still allow asymptotic freedom. In practice, however, this does not seem to work: There is no known examp le where all gluons

except for the photon have been given mass but asymptotic freedom remains .(2 6)

Another possible solution to this difficulty involves specific choices of relat ions among coupling constants , and is thus not natural. This possibility arises from the role of the Yukawa couplings in the renormalization group equations . Ordinarily Yukawa couplings don' t help , because they vanish at high energy faster than the gauge coupling; however for "eigenvalues" which determine the Yukawa couplings , the Yukawa couplings are proportional to the gauge coupling , and asymptotically free solutions can occur .(2 7) Attemp ts to use this possibility in realistic models lead to bizarre sets of quarks . For example

E. has given a model with _Ma( 2S)

p,n,\ 3* triplets c a,a' 3* triplets spin c 21 B 6* sextet c 3* triplets Higgs scalars . q,,¢' c } In addition the eigenvalue conditions must be re-adj usted in each order of perturbation theory .

c Hierarchies of symmetry breaking : The models with confinement can in ( ) some cases be unified at a mass scale of the order of the Planck mass. It seems likely that such a large mass is also necessary for complete uni fication in non-confinement models. < 29l

(d) There are many possibilities to choose a pattern of Higgs mul tiplets and a pattern of non-vanishing vacuum expectation values , so that mod els with }Q unbound color contain many more arbitrary parameters than the confinement models � )

61 4. THEORETICAL SUCCESSES FOR GAUGE MODELS WITH UNBOUND COLOR (30) (a) These models are theoretically consistent; in par ticular they avoid the U(9) problem by choosing commuting gauges for the strong and weak interactions prior to spontaneous symmetry breaking and diagonalization of the gauge boson mass matrix. In addition it has been shown that the desired spontaneous symmetry breaking pattern comes from a minimum of the Higgs potential, at least in lowest order. (b) Gauge models avoid large increases in leptoproduction structure functions above color threshold. In the naive Han-Nambu model, for example, where, as mentioned above, the photon has the form

- + under SU(3) x SU(3) , y (8,1) (1,8) f c the following increases in electron-proton and electron-neutron structure functions occur above color threshold (from quark contributions):

Structure function Increase above color threshold in naive Han-Nambu model

66 2/3%

100%

Similar effects occur for the neutrino-nucleon scattering. 2 Such large increases in leptoproduction structure functions at v and q in the present experimental range are experimentally ruled out . This fact alone eliminates unconfined color models where the weak and electromagnetic interactions are not gauged. < Gauge models avoid this problem in the following way. 30 ,3l) . The interaction Lagrangian can be written schematically as

where J is for weak) bas quark-flavor currents and lepton currents, and W (W Jc (c is for color) has quark-color currents and gluon-color currents. Before

62 spontaneous symmetry breaking , there is thus no lepton-color coupling . After

spontaneous synnnetry breaking this coupling would remain zero if the masses of

the gauge bosons are not split , since then the transformation to fields which

diagonalize the gauge-boson mass matrix would be an orthogonal transformation.

(Actually, in this case , the gauge-boson mass matrix would already be diagonal.)

Lepton-color coupling thus arises only due to mass splittings among the gauge

bosons . For neutral currents , the electromagnetic potential A which remains µ massless and the gauge boson which up to mixing effects transforms like the U µ color contribution to the electric charge, both give lepton-color coupling . The

lepton-color couplings from these two gauge fields must cancel exactly when the

fields have the same mass. When mass splitting occurs there is a partial

cancellation, expressed by

1 1 -2--2 2 2 2 2q q -Mu q (q -Mu )

in matrix elements, so that there is a suppression factor of

2 2 Mu 2 ( 2+Mu2 ) q Q

in cross sections relative to the cross section which would occur in a naive

(non-gauge) model in which only A mediates the lepton-color interaction . Thus , µ in gauge theory models , leptoproduction structure functions have the form

f lavor color color F + F F = O, below color threshold. i i i

Thus color excitation greatly suppressed . For Mu about 1 to 2 GeV , Pati and is Salam estimate 10 to 20% increases due to color excitation. This consideration has an important impact on the contribution of charged vector bosons , acting as

partons , in lep toproduction. In naive models, vector partons give a non-scaling

contribution to structure functions which varies as Q4 ; however in gauge models

63 they give a scaling contribution. The gauge models thus predict that oL/oT is non-vanishing and scales , an effect which for kinematic reasons is likely to show

up first at small x.

In addition to the suppression of color excitation because of cancellation

between photon-mediated and U-mediated interactions, there is a further kinematic

postponement of color excitation due to the fact that the masses of color-non-

singlet states are higher than those of color-singlet states . This postponement

can be described in a way similar to the corresponding effect for charm produc-

tion, wh ich is due to the higher mass of states with charm compared to the 32 charm-zero states . Several authors( ) have described this situation using a

new scaling variable 2 M c z x + M charmed hadron mass z_yy c

to replace the Bjorken x in the structure functions. This mechanism leads to

the postponement of the full color excitation to higher mass regions .

EXPERIMENTAL PROBLEMS 5 . a Colored gluons : A crucial feature of gauge models with color excitation ( ) 33 is the prediction of spin-one particles carrying color, the colored gluons ! ) If these particles have low mass, for example, less than GeV , we might expect 4 that they would have already been seen. The best place to look for the U, which + - + - couples to the e e channel, is in fonnation experiments. For e e colliding

vi.th total center-of-mass energy in the range 1.8 to GeV, Frascati, SPEAR and 7

1. 05 1. DESY experiments rule out U. From to 34 GeV, experiments at Novosibirsk + - (34) There is • 1T 1T an unexplored region between 1.34 and 1. 8 GeV where the U might be present. For l1U less than 1.1 GeV, the U is excluded, because if it were present it would lead to disagreement with g-2 for the muon.

If the colored gluons are very massive, say have mass mo re than 7 GeV, then

they might not have been seen; however , this possibility implies that effects

64 due to excitation of colored gluons are not related to present data; that is to

say various of the effects, such as dimuon production and possible quark decay cs an alternative interpretation of the events usually interpreted as a heavy

+ - lepton, seen at SPEAR and DESY in e , could not be connected with color- T, e excitation effects as desired by Pati and Salam. In addition , there is a plausible but not definitive argument against high-mass colored gluons : If colored-gluon exchange is the main interaction binding quarks into hadrons , then it is plausible that the hadronic size as revealed, for examp le , in electro- magne tic form factors is of the order of the Compton wavelength of the exchanged gluons(33). In this case colored gluons of mass greater than 7 GeV would produce ,

for example, a nucleon of much smaller size than the Compton wavelength p exp erimentally observed. (This argument is not defini tive because one could imagine that the characteristic size of hadrons might be determined by the colored gluon mass together with the masses of the color-singlet hadrons , and that light exchanges , i.e. and p-exchanges, are responsible for n- the relatively large size of the nucleon. ) + Clearly the best experiments to search for the U are e e- collision experiments . Nucleon-nucleon collisions which produce electron or muon pairs might also provide useful information. The latter experiments have the advantage that they scan a large range of possible U masses simultaneously, but the disadvantage that backgrounds make it difficult to rule out the presence of the U in a given mass range.

The octet of colored gluons contains four charged particles : -+ and v- p These charged colored gluons should be by their identifiable decay modes(1: 3)

-+ - - and + -+ ev ' BR 30% v- V e p i(*

-+ µ BR - 30% vµ ' hadrons in I=O (SU(3) = f 1) -+ hadrons in £ v £ or 1r. £ v K£v . I+ £ £

65 The most striking feature is the presence of leptonic , hadronic and semi-

leptonic fina l states , but the semileptonic final states with a single octet

meson are prohibited. This prohibition is because the gluons are flavor

singlets , and thus up to flavor-breaking effects they mus t decay into a

The leptonic decay mode of these par ticles is estimated flavo r singlet. to

) be of the order of for each of ev and (33 this large leptonic branching 30% e µv ; µ ratio gives rise to a large trimuon to dimuon ratio in neutrino experiments :

which is signi ficantly larger than the ratio expect ed from charm.

(b) Quarks . Pati, Salam and Sakakibara (3S) have suggested that the

events which have been int errup ted as corresponding to the heavy lepton, T,

in electron-positron annihilation experimen ts are really due to quark decays.

This interpretation requires the mass of the quark to be that of th e decaying

obj ect , i.e. GeV, and, in addition, it requires the gluon mass to be less 1. 9 than the quark mass . The decay mechanism wo uld be + - e e

The decay matrix elements have the form

- + V (1 + µ + µvq ] ( n I e ) v ) - (V Y5 ( Fly Fzo P)µ, i.e. gives a decay = +A if F This las t predic tion is in V 2 O. cont radiction with present data which indicates a V A coupling for the decaying - particle� 36) This interpretation also leads to the prediction of semileptonic anomalous

eµ even ts , i e .. , .

e e � e + -�- + + n +- n , etc. at a rat e es timated to be at about 2-5% of the anomalous eµ . Up to now no such semileptonic effects have been observed; however, present data does not

66 rule them out at this low rate. The quark decay interpretation also leads to

low energy mono energetic protons in the 10-50 MeV range near the eµ threshold

via the process

7) near eµ threshold? If what is called the T is really a quark , then it should

be seen in hadronic collisions at a rat e greatly exceeding the rate expected

from heavy lep ton production.

U. Sukhatme(38) has pointed out at this Rencontre that decays of un- + - confined quarks in e e annihilation should produce hadronic jets with con-

stant multiplicity, in contradict ion to exp eriments which show a mult ip licity

which grows with energy . This might be a problem for the unconfined-

quark interpretation of the jets.

Another possible effect of unbound color would be that of nucleon

dissociation to quarks via

in a high energy co llision. This leads to an ad ditional source of multi-

lep tonic final states . The possibility of proton decay also occurs in this

·· nodel ; however , e:-:r-:P""" '""', --sstve gluor..s are no t needed i the proton can be

kept sufficiently stable with gluons of only about 105 GeV, because in the Pati-

Salam model this decay is a 6th order process since each of the 3 quarks mus t

decay .

Several other Pati-Salam models are available, including models with

mirror symmetry, etc. However , I refer the interested reader to the literature

for discussion of these, as well as for further discussion of phenomenology ( 39 in these models. . )

7. SUMMARY OF TESTS TO DISTINGUISH CON FINED AND UNBOUND COLOR

� (a) Below color threshold As Chanowitz pointed out, since decays of low-

mas s mesons can be sens itive to the high-momentum behavior of integrals over

67 quark loops , such decays can, in principle, distinguish models with confined

° and unbo und color. The prototype is the decay n + 2y; however , although the

decay rate clearly disagrees with exp eriment in the model with a sing le

frac tional ly-charged quark triplet (or quartet , if cha rm is includ ed) , the

decay rat es are the same in both the model wi th three identical fractionally-

charged quark triplets with confined color, and in the Han-Nambu model with

three different integrally-charged quark tri plets with unbound color . These

latter two models do predict differen t decay rates for pseudoscalar mesons

and which are flavor-singlet and flavor-octet mixtures ; however , as (n n')

Chanowit z al so pointed out, uncertainties about the mixing angles as well as

the large mass extrapolations necessary in these cases make it difficult to

draw definit e conclus ions .

b) Charged co lor gluon-parton contributions give a non-vanishing

ratio in electroproduction which scales in gauge models with color o1/cT

exc itation. This test is not necessarily definitive, because similar

effects might arise as corrections to the confined quark-p arton model.

c) The neutral colored-octet glucn U0 which couples to leptons and + - can be formed in e e collisions must exist . The decay modes

+ hadrons + - 1 to 3 MeV U0 y, f

hadrons - 2 MeV + r + - _ 5 keV + i i r

for a of mass - 1.5 r,ev should allow this particle to recognized. U0

d) The charged colored-oct et g1uons + and + which can bes t be v� VK* + - produced in e e annihilation or in hadronic collis ions should have the branching rat ios

+ v- e BR - 30% e p v * -30% K + BR µv)J

3n , .•• BR - 30% + TITI , KK

£ BR - 50% TT7T V£ , KK.tv£

n£v1 , Kiv � i

68 e) Quarks will be uns table, decay via a V + A interaction, give semi

+ - leptonic µe ev ents in e e correspondin g to a 2-5% branching ratio , and

may give low energy monochroma tic gammas (10-50 MeV) corresponding to

radiative trans itions among quarks .

f) The nuc leon-to-quarks dissociation process contributes to µ µ

+ + events in vN co llisions , but does not contribute to µ µ events in vN

collis ions , because quarks decay to leptons only via

(n I)

and thus only produce negative leptons.

(g) The Pat i-Salam model predicts the following significant increases

4D) in neutral current neutrino cross sections above colo r thresho ld/

60-65% in a(�N) , 40-45% in a(vN) . These increases are reflected in the

following increases in and R , the neutr al- current to charged-current Rv v

cross section ratios : -10% for R- and -20% for R . Thes e effects are v v

rather specific to color excitation, becaus e flavor-changing neutral currents

are necessarily suppressed by the GIM mechanism.

I acknowledge discuss ions wi th Vic Elias , and a strong dissent from

the statemen ts made above by Jogesh Pa ti .

FOOTNOTES AND REFEREN CES .

· See W. Greenberg Nelson, Phys . Reports (to appear) , for a l 0. ana C. A.

review of models wi th confined as well as unbound color.

2. 0. W. Greenberg , Phy s. Rev. Lett. 13 (1964) 598.

3. The remark tha t the paraquark model can be cons idered a special case of

the three-triplet model seems first to appear in O. W. Greenberg and

D. Zwanziger , Phys . Rev. 150 (1966) 1177. R. Haag and coll abora tors ,

and A. B. Govorkov have studied the relation of thes e two models . See

Sec . 3 of Ref. 1 for a revi ew of the work of these latter authors and

for references to the literature.

69 4. M. Y. Han and Y. Nambu , Phys. Rev. 139 (1965) Bl006 ; Y. Nambu, in Pre

ludes in Theoretical Physics, ed. A. de Shalit, H. Feshbach and L. van

Hove (North-Holland, Ams terdam, 1966) , p. 133; see also A. Tavkhelidze,

in Seminar on High Energy Physics and Elementary Particles (International

Atomic Energy Agency , Vienna , 1965) , p. 763; P. G. 0. Freund, Phys. Lett .

15 (1965) 352; and Y. Miyam:i to, Prog. Theor. Phys . Supp. , extra number

(1965) 187.

5. The color terminology for the three-triplet model seems firs t to appear

in D. B. Lichtenberg, Uni tary Symmetry and Elementary Particles (Academic ,

New York, 1970) , pp. 227-228; however th e popular use of this terminology

was st imulated by the surveys of applications of the identical three

triplet model to hadron physics given by Gell-Mann and collaborators ,

M. Gell-Mann, Acta Phys. Aus triaca Supp . 9 (1972) 733; W. A. Bardeen,

H. Fritzsch and M. Gell-Mann , in Scale and Conformal Symmetry in Hadron

Physics, ed. R. Ga tto (Wiley , New York , 1973) , p. 139; M. Gell-Mann , in

Proc. XVI International Conference on High Energy Physi cs , ed. J. D. Jackson

and A. Roberts (NAL , Batavia, 1973) , Vol . 4, p. 333.

6. This equivalence was pointed out in 0. W. Greenberg, University of

Maryland Technical Report 680 (1967) (unpublished) , and in 0. W. Greenberg

and Resnikoff, Phys . Rev. 163 (1968) 1844 . M. 7. Gurs ey and L. A. Radicati , Phy s. Rev. Lett. 13 (1964) 173. F. 8. R. Horgan and R. H. Dalitz , Nuc l. Phys. B66 (1973) 135, erratum 'B71 (1974)

54 6; R. Horgan, Nucl . Phys . B71 (1974) 514 ; R. H. Dalit z, in New Directions

in Hadron Spectroscopy, ed. S. L. Kramer and E. L. Berger (Argonne National

Lab . HEP-CP-75-58, 1975) , p. 383 ; R. R. Horgan, J. Phys. G: Nucl. Phys .

(U .K.) 2 (1976) 625 and Proc. Top. Conf. on Baryon Resonances , Oxford, 1976 .

9. M. Jones , R. H. Da litz and R. Horgan, Rutgers-Oxford preprint (preliminary

version) , July , 1976.

10 . R. Cutkosl')' and R. E. Hendrick , Carnegie-Mellon preprints C00- 3066-81 and

-84 (1976) .

70 11 . A. DeRuj ula , H. Georgi and S. L. Glashow, Phys. Rev. D 12 (1975) 147.

12 . J. F. Gunion and R. S. Willey , Phys . Rev . Dl2 (1975) 174 .

13. A. Chodos , R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf,

Phys. Rev. D9 (1974) 3471; A. Chodos , R. L. Jaffe, K. Johnson and C. B.

Thorn, Phys . Rev. D 10 (1974) 2599; T. DeGrand, R. L. Jaffe, K. Johnson and J. Kiskis , Phys. Rev. D 12 (1975) 2060.

14. C. Rebbi , Phys. Rev. D 14 (1976) 2362

15. R. L. Jaffe, Phys. Rev . D 15 (1977) 267 and 281, and Phys. Rev. Lett. 38

(19 7 7) 195.

16. R. H. Dalitz, Proc . Topical Conference on Baryon Resonances, Oxford,

1976 (to appear) .

17. I. H. Dunbar, Oxford prep rint 94/76.

18. D. B. Lichtenberg, Lett. Nuovo Cimento 13 (1975) 34 6; A. W. Hendry and

D. B. Lichtenberg, Phys. Rev. D 12 (1975) 2756.

19. C. A. Nelson, Phys. Rev. D 13 (1976) 2139.

20. See Re f. 1, Sec. 4 for further discussion and more references concerning

saturation.

21. G. S. LaRue , W. M. Fairbank and A. F. Hebard, Phys. Rev. Lett. 38 (1977) 1011

have recently announced the observa tion of fractional charge on smal l

niobium balls. Confirmation of this observation would change greatly

our present viewpoint.

22. Y. Nambu and M.-Y. Han, Phys . Rev. D 10 (1974) 674.

23 . J. C. Pati and A. Salam, Phys. Rev. Lett. 31 (1973) 06 61; A. D. Sakharov, JETP Lett. 5 (1967) 24.

24. It is amusing that the niobium balls of Ref. 22 have approximately the

Planck mas s.

25 . J. C. Pati , Phys . Rev. D4 (1971) 2143; J, C. Pati and A. Salam, Tries te

Internal Report IC/73/81; and A. D. Dolgov, L. B. Okun and V. I.

Zacharov, Phys. Lett . 47B (1973) 258 .

71 26. D. Po litzer, Phys . Reports 14 (1974) 129; T. P. Cheng , E. Eichten and H.

L. F. Li, Phys . Rev. D 9 (1974) 2259.

27. E. S. Fradkin and Kalashnikov, Phys . Lett. 59B (1975) 159; N. -P. 0. K.

Ch ang, Phys . Rev. D 10 (1974) 2706.

28 . E. �!a , Phys . Lett . 62B (1976) 347 and Univ. of Oregon preprint OITS -64

(1976) .

29. V. Elias , Phys. Rev. D (to appear) .

30. J. C. Pati and A. Salam, Phys . Rev . D 8 (1973) 1240 ; later work reviewed

by J. C. Pa ti , in Gauge Theories and Modern Field Theory , ed. R. Arnowitt

and P. Nath (M.I.T. , Cambridge, 1976) , p. 27.

31 . C. Pati and A. Salam, Phys. Rev . Lett . 36 (1 976) 11; G. Raj asekaran J.

and P. Roy , Pram::;na 5 (1975) 30 3, Phys. Rev. Lett. 36 (1976) 355, and

erratum , ibid 36 (1976) 689.

32. A. DeRuj ula, H. Georgi , S. L. Glashow and H. R. Quinn, Rev. Mo d. Phys.

46 (1974) 391; H. Georgi and H. D. Politzer, Phys. Rev . Lett. 36 (1976)

1281 and 37 (1976) 68 erratur.: ; R. M. Barnett, Phys . Rev. Lett. 36 (1976)

1163 .

33. J. C. Pat� J. Sucher and C. H. Wo o, Phys. Rev . D 15 (1977) 147.

34 . V. A. Sidorov, Proc. XVIII Int . Conf. on High Energy Physics , Tbilisi,

1976 (Dubna, 1977) Vol . II, page Bl3 . R. F. Schwi tters , op cit, page

B34. S. Bartalucci, S. Bertolucci, C. Bradaschia , et al, Frascati pre-

print INFN/AE-76/10, report evidence for a vector meson at 1.1 GeV from

+ - yp -+ e e p. This evidence seems unconvincing because it requires a

subtrac tion to isolate Bethe-Heitler--Compton interference in wh ich

the effect appears.

35 . J. C. Pati , A. Salam and S. Sakakibara, Phys . Rev. Lett. 36 (1976) 1229.

36. M. Perl, SLAC-PUB-1923 , Apri l, 1977 gave th�

2 following x for fits to the data for the number of e's or µ's vs.

normalized momentum r � (p - 0. 65 GeV/c)/(P - 0.65 GeV/c) : max

72 Fit: V-A V+A V+A with lowest-r point omitted : 60% <0.1% 5% x 2 The neutrinos are assumed above to have zero mass.

37. D. Coyne, Proc . XVIII Int. Conf. on High Energy Physics, Tbilisi, 1976 (Dubna, 1977) Vol. II, page N66 , reported detection of a mono-energetic photon line..'55 MeV at the Tbilisi Conference. This line might come from transitions among D's or am:ing F's, as well as from quark decay. 38. U. Sukhatme, private communication.

39. The literature on Pati-Salam models can be traced from J. C. Pati, University of Maryland Technical Report No. 77-073 (1977) , Lectures at Scottish University Sumner School, St. Andrews, August, 1976.

40. R. N. Mohapatra, J. C. Pati and D. P. Sidhu, Phys. Rev. Dl5 (1977) 1240.

73 74 HADRONIC \HDTHS IN CHARMONIUH

J.D. Jackson" ) CERN , Geneva , Swi tzerland

Ab stract: Theoretical estimates for che hadronic widths of the s-wave and p-wave states in charmonium .(based on lowest order perturbation theory in QCD) are compared with indirect inferences from experiment. For the p-states, the expected ratios of 15:� 1:4 fur JPC = l ++ 2++ , respective!y, ++ are in �ccord with the present, rather crudeo exper, ime, ntal evidence, and the theoretical l y less reliable ab solute magnitudes are within a factor

of three or four. For the pseudoscalar = 0) states the situation is (2 less clear. Identification of the as n� causes difficulties for (/CD . X(3454)

R�sum� On compare aux indications exp�rimentalcs indirectes les estimations th€oriques des largeurs hadroniques des �tats du charmonium de moment

angulaire orbital = 0 et - 1. Pour • 1, les rapports calcul§s i 9 C Q sont de pour JP = ++ ++ ++ respectivemen t, ce qui est 15:' o , 1 , z en accord avec1:4 ]es donn�es exp�rimentales , d'ailletJ rs assez grossi�res . Les valeurs ab snlues de ces largeurs , pour lesque lles le calcul th�orique est moins -sGr , sont correctes un facteur trois ou quatre pri?s . La 5 si tuation des 6tats pseudos calaires n'est pas aussi claire. L'identifi cation de l'� tat avec n� cause quelques difficult§s la chromo dynamique quant iquey(3. 4��) �

leave frorn the Uni vers ity of California, Berke ley , Calif. , USA . ·k ) On s

75 1. INTRODUCTION

The presently known states of charmonium below the charm threshold of approxi-

mately 3.8 GeV are shown in Fig. 1. The two J = states , w(3095) and w'(3684) , 1 are well established in all their quantum numbers . The existence of the three states called x(3414) , X(3508) and X(3552) is beyond doubt; the assignments PC = J O++ , l++ , 2++ are consistent with all observations and are more or less strongly implied1). The remaining two states , X(2.83) and x(3454) , have been more 2 firmly established recently ' 3), but so far have no quantum numbers assigned,

= apart from C +l. The number of states and the rough ordering are just what is expected from a confined posi tronium-like spectrum, with n = and n = 2 1 S and 1 0 3S , plus n = 3P states (the 1 Pi 'state is reachable from the w' only via two 1 J photon emission and is therefore expected to be seen only with difficulty) . We adopt this interpretation throughout. Only in discuss ing the X(3454) do we mention another possibility, still within the "positronium" framework .

3 5 w' (3684) 1

n = 2 x (3552) 3 p 3 2 X(3508) p l ( 1 S ) ( 3454) o x 3 po x0414)

w (309 5)

= n

X(2.83) <1so)

Fig. l Energy levels of charmonium

The basic properties of the spectrum, the size of the wave functions , and the magnitude of matrix elements not involving spin-flip can all be understood in terms of almost anyone 's non-relativistic Schrodinger-equation description of a

massive charmed quark-antiquark pair (cc) . Re lativis tic corrections (Fermi hyper fine interaction, tensor force , etc. ) are less satisfactorily given , with the ob served 3Si-1 So splitting being a factor of three to five larger than expected, but that is another subj ect.

76 Our concern is the annihi lation of these states into ordinary hadrons within the framework of QCD , a non-Abelian gauge theory of flavourful quarks interacting via colourful, massless , vector gluons . As is well known , such a theory possesses "asymp totic freedom" , that is, the property that the effective coupling cons tant g(p) becomes smaller, the larger is the energy or momentum scale (p) being con sidered. By uncertainty principle arguments this leads to the hope that at short

distances of separation of the quarks the coupling cons tant = g2 /4n will be s small enough to permit the use of perturbation theory . This a hope is the basis of the terminology "charmonium", and the discuss ion of annihilation of the new part icles into ordinary hadrons in analogy to the annihilation of positronium into photons .

Very much in the same spirit as the calculation of the expected value of

R = o (e+ e - hadrons)/o (e+ e - +-) from the materialization of the virtual + OCD + µ µ photon into a quark-antlquark pair, without inquiry into the messy details of how the quarks manage to convert themselves into ordinary hadrons , the annihilation of a (cc) bound state is estimated in QCD by calculating the rate of transformation into the minimum number of gluons permi tted by selection rules . The assumption is that the gluons are sufficiently close to their mass shell (even though they never escape) that the subsequent conversion into ordinary hadrons occurs with unit probability . This is certainly an act of faith that can be challenged, but the plateaux seen in R as a function of energy indicate that this basic philosophy ' ) may not lead to gross error in estimating total rates of annihilation into hadrons .

2. S-STATE ANNIHILATION

For 150 and 351 positronium states , the lowest order annihilation is into two ) and three photons , respectively . The rates are 5

(1) re 50 �){�)= 16 2 2 1f (2) '1 9 ) r\J""�n) 1f

where 1/137, Mis the mass of the decaying sys tem and R(r) is the s-wave radiala wave= function of the bound state . For bound states of quark-antiquark these decay rates into mus t be mu ltiplied by a colour factor of three and photons a factor of eQ for each power of a.

77 Another rate of interest is the annihilation of a massive charged fermion - antifermion bound state into e+e . For quarks with colour, this rate is

( 3)

(1) The transcription of the above positronium formulae, and (2) , to qq �

� gluons involves certain factors arising from the non-Abelian nature of QCD, a namely the traces of the product of n SU(3) matrices (\ /2). For n = 2, the

2 3 5 18 2 recipe is a � 2a�/3; for n 3 it is a � a�/ . Here a g /4n is the QCD = s = "fine structure" constant. For a bound 35·1 cc state such as 1)1(3095) , the direct annihilation into hadrons is thus assumed to be given by

The coupling of to e+e- is given by Eq. (3) . The ratio of Eqs . (4) and (3) , 1)1 experimentally determined to be 10 ± 1.5 is l), (� - r::f°e

= 0. 187 �$ ±0.10

The error in as is just from the experimental error in the widths -- the systematic

(theoretical) error is unknown ! We can be gratified that a 0.2 for systems of M" 3+ GeV. So far, nothing is proved about QCD , but at leass t is has not been shown to be patently ridiculous . In passing we note that we can now estimate various rates for the pseudoscalar partner of the 1)1, and also, from the ob served ratio of re 's for 1)11 and 1)1, for the 1)11(3684) and its pseudoscalar partner:

78 (8)

P stand for the pseudoscalar and vector partners i In Eqs . (6) and (7) , and V (nc , jJ or From Eq. (8) and the observed r (ijJ' ) 2.1 keV, we conclude that n�,iJ!'). e = ' 9% rdirect(ijJ + h) "' 21 keV, implying that only 21/228 "' of iJ!' decays go directly into ordinary hadrons . This is quite consistent with the book-keeping of the various decays 1) . ij!1 Identifying as n Eqs . and give yy keV, X(2.83) c ' (6) (7) f(X + ) "' 6-8 ft (X) "' 4.7-5.7 MeV, depending on whether one takes IRp (O) /Mp I = IRv (O) /M v I or only IR (0) I = IR (0) I. The value 1/738 1.36 10-3 for the branching ratio of p v = x yy is a firmer QCD expectation. For the the widths are reduced by a nc + n� , factor of � 0.44, modulo ratios of masses squared, from fe (ijJ 1)/fe (ijJ) . The present

experimental situation on these widths for n and are discussed below. c n�

3. P-STATE ANNIHILATION IN QCD

An opportunity for a test of the ideas of QCD , rather than merely a determina 6 7 ) tion of parameters, is provided by the hadronic widths of the p-states ' [x (3414) , ++ ++ X(3508) , and X(3552) of Fig. l] . For the 3Po (O ) and 3P2(2 ) states, the lowest order annihilation is into two gluons , as indicated in Fig. 2a, while for the ++ +- 3P1 (l ) and the 1P1 (1 ), the three-gluon or single-gluon plus light qq pair pro cesses of Fig. 2b presumably dominate. Since the even J rates are proportional

to while the J = rates are proportional to and we expect a� , 1 a! , as "' 0.2, X(3508) to show up more strongly in the cascade transitions , Y1X, X + Y2iJ!, ij!1 + and less strongly in the transitions , + y1x, X +hadrons , compared with x<.3414) ij!1 ) and x(3552) . This is indeed what is observed1 , and forms a qualitative success for the ideas of QCD .

c (a)

(b)

a) Two-gluon annihilation for 3P0 and 3P2, b) Three-gluon or Fig. 2 qq-gluon annihilation of 3P1 and 1P 1

79 ) The test can be made more quantitative6 '7 . Firs t of all, the relative values

of the hadronic widths of the p-states can be computed , Then, with less reliability, the absolute magnitudes can be determined from the properties of the bound-state wave functions. We sketch the basic idea for the two-gluon annihilation of the O++ and 2++ states . For an exothermic reaction proceeding via the s-wave , it is well known that near threshold in the entrance channel the cross-section behaves as

where v is the relative velocity ( flux factor) and the cons tant is the transition probability at threshold, for normalization of one particle per unit volume in the incident beam. For higher partial waves , barring some peculiarity, one has

· OJ_ = � -t2tl" ( consto.nt") where p is the c. m,s. momentum in the entrance channel. For the same process ) proceeding from a weakly bound state of the incident pair, it can be shown 5 that the transition probability is given in firs t approximation by

(9) 1 im c.-e. l'.J"-+ 0 [ l\Jca]-12-

where S is a statistical factor that may occur because the spin and orbital angular population of the incident beam may differ from that of the bound state in ques tion, C is a numerical factor, and R (r) is the unperturbed radial wave function of the £ £ bound state of angular momentum £. For s-waves , £ = 0, = 1/41; , and = C o r lim(va0 ) [w(O) [ 2 , the familiar result that led to Eqs. (1) to (3) . V+O The procedure is thus to compute G (p) , isolate the p 2£ dependence at thresh £ old, and hence find S(vG /p2£). The relative values for different J (but the same £

+ £) will then give the ratios of widths for the exothermic process. For 3PJ gg , 1 we have £= and J = o++. 2++ For the collision process, cC � g 1 g2 , we examine the helicity amp litude (A1 A2 [.At,[%,%). Calculation shows that at threshold, the two independent amplitudes are

A -c for 1 (10 ) for A { + C 1 �= sin.�-& where 8 is the angte of the momentum of one gluon relative to the beam direction .

Since helicity amp litudes have the angular variation d� (8) , A=A1 A2 , µ = J J µ - = A - A- we expect doo(8) for A = and d 0(8) for A = -A • Inspection of a c c ' 1 A2 2 1 2 J J = 0 = = 2 table of '\ii.i• shows that the matrix element (10) has for A 1 A2 and J

80 We can now compute the ratio of widths for the ++ and O++ states to annihilate 2 into two gluons . There are two factors : (i) the integration over angles in the final state , (ii) the different probabilities of the incident beam of cc to have

J = 0 and J = 2:

Final state ratio

The initial state of the cc plane wave with A L and 1, 1 has a c c 9, s spin-angular wave function,

= -J,( :J = o) +ff( J z)

J = 2 J = 0 2:1. For pure = 2 The ratio of to in the incident beam is thus J or pure = bound states , we must therefore replace the "collision" matrix ele J 0 2 ments (10) by and sin for J = and J = 2, respectively. Th is in 13c -1372c 8 0 troduces a relative factor of in the ratio of widths . Thus we obtain finally , 1/2

4 _ 8 1 ! (2++ _ ��) --- �-- - (11) 1 ( o++_,.d� ) 15 2 1.S

This result is fairly specific to QCD , with two massless vector gluons in the final state . If, for examp le, the gluons were scalar, the matrix element (10) would be replaced by the single amp litude , C (cos2 -2) . This is a mixture of J = 0 s 8 and = Legendre polynomials in the ratio 5:2 in amplitude , leading to a final J 2 state ratio of

= + 125

Scalar gluons would thus lead to 2/125 for the ratio in Eq . (11) .

Keeping track of all the factors leads to the ab solute rate for the O++ p- state 6) ,

(12)

81 where R1(0) is the derivative of the p-state radial wave function, evaluated at the origin. For the = states ( 3P1 and 1P1), annihilation proceeds as in Fig. 2b . J 1 Here things are somewhat trickier than for the even J states . For both ggg and g(qq) final states , there occurs a singularity at zero binding, i.e. the matrix element for the collision, divided by p, is not analytic for p -+ O. Our simple 7) argument must be augmented by more elaborate considerations . An approximate result for the annihilation rate is

(13) where

12..8/311' = J3. ' N= { 320/91r = 11.3 3 [g(qq) dominates the P1 rate, while ggg dominates 1P1]. The presence of the logarithm is a signal that Eq. (13) is more approximate than Eq. even granting (12)-, all the QCD assumptions . Terms of order one relative to the logarithm have been neglected. The logarithm can be estimated in a number of different ways. One is to say that the binding energy is of the order of a�mc. Then

Another is to say that the binding energy is the difference between the charm threshold of � 3.75 GeV and the mass of GeV. This gives The ratio Xi 3.51 2.1. of J = annihilation into hadrons to J = annihilation into hadrons is thus , l++ 2++ from Eqs . and (13) , (12) 13,, ol5Jrn( ) f'(i+� h.) o . .10!,,. .. ( ) � o.3 .r(2+� �) � . q6 The QCD prediction for the relative values of the hadronic widths of the triplet p-states of charmonium is therefore

+ (14)

Much less reliable are the actual values of the hadronic widths from Eqs . and (12) With a = 0.187 and the favourite wave functions of Ref. 6, one finds (14) . s

82 ) (15) o++ -::: 2..2. M V .l;._r;: ( e Other plausible estimates of the derivative of the p-state wave function at

= 0 ' r give values from 1.4 to 1.8 MeV . Since IR (O) 1 2 depends on the inverse length scale parameter � as �5, and is also sensitive to the detailed shape of the potential, the value (15) should be taken only as indicative of the general order of magnitude.

4. COMPARISON WITH EXPERIMENTS FOR THE P-STATE We are now ready for a confrontation with experiment. There are data on the 3 8) branching ratios B1 for + 1 for the 3P final states ' and also on the ¢' Y XJ J ) products for the cascade + 3 , as shown in Fig. 3. From B1B2 ¢' + Y1XJ Y1Y2¢ these data, the branching ratio B2(J) for + can be deduced for each of the XJ Y2¢ three 3P states .

7.5 ± 2.6% (Bi ) 3% 8 ± 3% I 9 ± 3%

2.4 0.2I, .. ±0 .8% ±0 . 2.%

Fi g. 3 (Top) Branching ratios B1 in per cent for the radiative transitions, ¢' + + (Bottom) Product of branching ratios B1B2 in per cent for the Y1XJ· + cascade transitions , ¢ Y1XJ + Y1Y2¢ Assuming that the only significant competing processes are + y and + XJ 2¢ XJ +hadrons , the width into hadrons can be expressed as rh. (J) =(s�(:J)- i)�:i.( :r) If the radiative width were known, we would have an experimental determination of Unfortunately, we have no direct information on The next best fh (J). fy (J). thing is to use some plausible estimate of the relative values2 of for dif fy (J) ferent J and thereby estimate the relative hadronic widths for compar2ison with Eq. (14)9) . Specifically, we have

83 (16)

The trans itions � Y1X and X � Y2iJJ are plaus ibly predominantly electric dipole lj!1 J J transitions . For a group of initial states in the same Russell-Saunders CxJ l mu ltiplet, the dipole matrix elements can be taken as approximately equal . Then

the relative rates for a radiative transition to a common final state are just 3 3 proportional to k • With this k assump tion and the data shown in Fig. the 3, numb ers are

We thus find 1 = 3.+

for comparison with Eq . (14) . The agreement is quite acceptable, cons idering the

poor statistics on the experimental values of B1B2 • Note , in particular , that for ++ J = O = ) , B1B2 0:2 ± 0.2% (based on one event . The ratio is therefore 00 6.3:1 + real ly some thing like _ 3 • 2) Clearly , better data are needed before a (6 .3 :1: ++ ++ strict tes t can be envisioned . The scalar gluon ratio of 125/2 for 0 ,2 cannot

even be excluded at the moment .

To test the ab solute magnitudes of the hadronic widths it is necessary to have (J) . One estimate can be made by using some bound-state model. If such f y, a mode l gives reasonable values for the ab solute radiative rates � Y 1 X ' it is �' J plaus ible to trus t its predictions for the second transition. Even cruder is to

take a harmonic oscillator estimate, reduced by a factor of two to account for

the distribution of the oscillation strength over several levels rather than just

one 10) Th is gives � 100 keV , 230 keV and 300 keV for = 0, 2, respec ry, (J) J 1, tively . Then one finds f (J) � MeV , 0 . 6 MeV , 2 MeV, 1n the same ordering . The h 4 corresponding QCD values, from Eqs . (14) and are 2.2 MeV , 0.15 MeV , 0.6 l!eV . (15) ,

84 Another me thod estimating the magni tude of f (J) is by the use of dipole of y2 1 1 ) sum rules . With some assump tions about the underlying dynamics of the bound

states , these give upper and lower bounds on f (J) . The ranges are found to be y 2 160-240 keV , 230-400 keV , and 280-480 keV , for J = 0, 1, 2, respectively, yielding

hadronic widths f (J) � 6 9 MeV , 0.6-1 .1 MeV , 2.0-3.4 MeV . These are somewhat h - 1 2 ) larger than the cruder estimate, but not much .

3 For the P states the situation at present is that the re lative hadronic J widths seem in accord with the expectations of QCD , although the data are subject to rather large uncertainties at present . The absolute magnitudes , inferred some

what indirectly from experiment, are a factor of perhaps four larger than the

rather uncertain QCD estimates .

5. THE PSEUDOSCALARS

We now turn to the alleged pseudoscalar states , X(2.83) and X(3454) . First,

the X(2.83) . The experimental observation of this state is via three photons in 2 ) the cascade, � y1X, X + The observed branching ratio product is (1.2 ± -4w Y 2Yi· x ± 0.5) 10 • On the other hand , the radiative transition Y1X has not been 3 W 8 +) seen at SPEAR in the inclusive photon spectrum from decay ' . The upper limit W on the branching ratio is 3% . Thus we have the experimental inequality,

-4- L 2 .to -3 > = 4 .10

- 3 to compare with 1/738 = 1.36 x 10 from Eq. (6) . Since the error on the 4 is

±1.7, there is no immediate cause for alarm . If the transition yX is not w + seen at the 1% level, however, there will be an order of magnitude discrepancy 13) between a fairly firm QCD prediction and experiment . Something will then have to give !

The situation with respect to the identification of the xC3454) as the n� is even less satisfactory . No hadronic decays have been observed so far. It is 3 ) only seen in the cascade , y1x, y2w with a product of branching ratios , w '+ X + - 2 = x • B1B2 (0.8 ± 0.4) 10 An upper limit of 5% can be set for 8 1 from the in 8 ) clusive photon spectrum from decay . This means that B2 0.16 0.08, cor w' > ± responding to a large radiative width or a small hadronic width . There are various ways of getting a handle on the magni tude of the hadronic width . The most na1ve

approach would be to say that both radiative tran�itions are allowe d magne tic di pole transitions with the same matrix element . Then the radiative widths are in 3 the ratio, /f 3(0.340/0 . 223) 10 .7. From the upper limi t, f < 0.05 x f y 1

x = Y2 Yi x � 228 11.4 keV , we infer rt < 10 .7 11.4 keV/0 .16 0.76 MeV . Th is is a factor of three smaller than the QCD expectation of ab out 2.2 MeV [see below Eq . (8)] .

85 Ano ther and more intelligent way of proceeding is to observe that if the X(3454) is the partner of the �1(3684) we expect their spatial wave functions to be similar and so the first transition, �· + Y 1 X, to be an allowed Ml transition, with a radial overlap integral of roughly unity. The es timated rate is 18 keV , only a factor of two larger than the upper bound of 11 keV. But the second tran sition is another story. Since the �· and n' are rad ial excitations , their spatial wave funct ions are largely orthogonal to the ground state wave function of the

�(3095) . The transition, X + y2�, is thus an unfavoured Ml with a very small overlap integral . The unobserved transition, �· + yX(2.83) , is a similarly un favoured transition. In fact, from the upper limit3 •8) of 1.1% for �· + yX(2.83), corresponding to a width of less than 2.5 keV , one can expect the width for + X + y2� to be no more than 3 2.5 = 7.5 keV 1 4 ) Then we conclude r < 7.5/0.16 x t 50 keV .

Other ways of es timating or setting a bound on r lead to numbers nearer the y2 50 keV figure for r than the over-simple 760 keV. Thus there appears to be a t fairly serious discrepancy (by an order of magnitude or more) between QCD predic tions and "experiment" if the X (3454) is identified with the n�.

1 PC An alternative identification of X(3454) as the D2 -state (J 2 -+ ) has been suggested1 5 ) , This at first seems unpalatable, but in view of the apparently large "hyperfine" splitting between the X(2.83) and �(3095) cannot be excluded . The triplet d-states are expected just above the �'; the singlet d-state could be depressed to 3454 MeV. The radiative transitions through the 1 D2 -state are both unfavoured Ml 's, involving spin-flip and an over lap of s- and d-state radial 3 wave functions , if the � and �' are assumed to be pureiy 5 1 -states. The expected transition rate for y x is so small as to be incons istent with the observed 1µ ' + 1 - branching ratio of B B = (0.8 0.4) 10 2 This 1 D interpretation can only hope 1 2 ± x • 2 3 to make sense if there is an admixture of D1 in the � and �·. This is quite plausible on general grounds, QCD having in it a tensor force contribution to the binding potential . With the simple quark model estimate5 ) for allowed Ml transi tions , one finds r z 18 E� keV , r z 38 E keV, where E and E are the intensity y1 y2 D D � fractions of d-states present in the � and �' . The upper limit of < 11 keV rY 1 > sets the bound , E� < 0.6. The lower limit, B2 0.16 ± 0.08, implies 2 f < (240+ 40)E keV. The percentage of d-state in the � is probably quite small t - a o D (ED : 0.01) , leading to an estimated total width of a few keV or less. The 1 expected width of the D2 state is small, but unlikely to be that tiny1 6). Thus 1 the D2 hypothesis, while not completely excluded , has its own difficulties . Besides , such an assignment means that the n� is still to be discovered ! One problem has been replaced by another .

86 Added remark (in answer to a question of Jacob) M. In estimating the hadronic widths in QCD one computes only the lowest order annihilation into gluons , treated as real massless vector particles with only two transverse states of polarization. Might not these estimates be modified drastic ally by the inclusion of the longitudinal polarization state of a gluon by making it off mass shell by coupling to a light qq pair? Admittedly, there is another power of as present, but perhaps the longitudinal contribution is so large as to cancel out the factor of a 0.2. s � The answer is that for an allowed process like n gg the addition of the c + Dalitz pair contributions only contributes a small correction (ignoring the ques tion of double counting) . Specifically, the Dalitz-Kroll-Wada formula for the relative addition is

where is the mass of the decaying state and m is the light quark mass. With 11 M = 3.5 GeV, m 0.34 GeV and a = 0.187, we have 0.05. Omitting the -7/2 s p = term raises this to 0.20, still a small correction. p �

87 REFERENCES AND FOOTNOTES

For the experimental facts and inferences from them, see G.J. Feldman , � spec 1) troscopy , Proc. 1976 Summer Ins titute on Particle Phys ics , Stanford Linear Accelerator Center, SLAC-PUB-1851 (1976) and SLAC Report No . 198 (1976) , or B.H. Wiik and G. Wo lf, Electron-posi tron interactions , Les Hauches Summer Sch ool , 1976, DESY 77/01 (1977) .

2) W. Braunschweig et al., DESY 77/02 (1977) .

3) J.S. Whitaker et al., Phys . Rev . Letters 37, 1596 (1976) . The three or four events in this experiment �' -+y 1 y(34511) , x(3454) -+ y2� re until seen as we , recently , the only evidence for the y(3454) . At this conference , V. Blobe l has repurted three events from PLUTO .

Critics point out tl1at wl1ile the annihilation of a massive state may few-gluon 4) involve small coup ling cons tant for each tl1e many-gluon annihila � gluon , tion presumab ly has a large coupling cons tant for each , because in sucl1 a configuration each gluon is 11s oft11 • Hhy , they ask, can one then believe the lowes t-order calculation? I 11ave no solid answer for tl1is , but simp le minded �stirnates for many-gluon rates indicate that if the effective cou- p ling constant for each of gluuns does not grow with N too rapidly, for N examp le O /Sl2 N/2, the sum of annihilations into an arbi trary number of N '\_, gluons differs by only a modest numerica] factor f rom the rate into the smallest number of gluons , at least for the charmoniurn value of 0.2. Sl2 ::--

For an elementary discussion of the calcu lation of such rates , see 5) J.D. Jackson, Lectures on the new particles, Proc. 1976 Summer Ins titute on Particle Pl1ys ics , Stanford Linear Accelerator Center, Report No . 198 (1976) .

6) R. Barbieri, R. Gatto and R. Kogerler, Phys . Letters 60B, 183 (1976) .

7) R. Barbieri , R. Gatto and E. Remiddi, Phys . Letters �. 465 (1976) .

8) D.H. Hadtke et al. , Contribution to the 18th Internet. Con£ . on High-Energy Physics, Tb ilisi, July 1976.

9) M.S. Chanowitz and F.J. Gilman, Letters �. 178 (1976) . Phys . 10) This is discussed in Re f . Such estimates agree well enough with model 5. calculations , for example E. Eich ten et al ., Phys . Rev . Letters �. 369 (1975) ; 500 (1976) ; K.M. Lane (private comm nication) . l_;)_ , u 11) J.D. Jackson , Phys . Rev . Letters E· 1107 (1976) .

12) The spread in values comes solely from the upper and lower bounds from the sum rules . The errors in the branching ratios have not been included .

13) There is already a discrepancy of a factor of perhaps 10-16 between the ex pected rate for �-+yX and the 3% upper limit of 2.1 keV . Ml

14) There is . in fact, a rapid energy denendence to the width of an unfavoured Ml transition. Naively, it is as k7 • This would reduce the estimate for drastically. X + y2� 15) H. llarari , Phys . Letters 64B, 469 (1976) .

16) We can appeal to Eq . (9) and estimate very crudely that r (1D2) : (2[,/M) x h 2 x where �� 0.5 GeV is the scale parameter of the oscillator wave r ( 3 P2), funch t ions that fit the energy leve l spacing and electric dipole momen ts . This gives � 50 keV , an admi ttedly very rough estimate . fh ( 1 D,) 88 PARTON MODEL STRUCTURE FROM A CONFINING THEORY

J. Kripfganz CERN , Geneva , Swi tzerland

Ab stract Lepton pair proouct1on in hadronic scattering is studied in QCD in l+ l dimens ions . From this confinement model the Drell-Yan parton model result is recovered.

Resume On etudie la production des paires de leptons dans le modele de QCD a l+ l dimens ions . On recupere le resultat du modele des partons de Drell et Yan.

89 Quantum chromodynamics (QCD) -- the colour gauge theory of quarks and gluons - is a promising candidate for a theory of strong interactions. This has been empha sized in many contributions to this meeting. QCD has the important property of being asymptotically free. This allows predictions for a few quantities which are determined by the short-distance or light-cone structure of the theory (like 2 CTtot e+e- or the Q dependence of the moments of the structure functions). More detailed predictions are made by quark-parton models. Models of this kind are phenomenologically quite successful. Applying these models naively, however, one might expect the appearance of quarks in the final states of the corresponding reactions . This is not observed. The hope is that the long-distance structure of QCD confines quarks and gluons such that only colour singlet states appear in the spectrum of the theory . But can this be arranged without destroying the parton model structure where it works quite well? We have studied1) this question recently for lepton pair production in hadronic 2 scattering, a process which is not light-cone dominated ) . A confinement model with many realistic features which is appropriate for this purpose is QCD in l+l dimen -6 sions 3 ) . This model can be explicitly solved1 ' 3 ) in the limit

N c -.. oo where SU(Nc) is the colour gauge group and g is the coupling constant of the theory. This limit corresponds to the infinite sum of all planar diagrams without internal fermion loops. Confinement is more or less trivial in two dimensions because of the linearly rising gluon potential. Correspondingly, the spectrum of the theory is discrete 3 up to infinity and contains no qq continuum ) . Since the theory is super-renormalizable it behaves like a free field theory in the light-cone limit. There are no logarithmic scaling violations . The inter esting problem to work out is how the free-field parton-model results can be ob tained from the theory without free quarks . For e+e- annihilation and deep inelas tic scattering, this has been studied in Refs . 4 and 5. Let us now consider lepton pair production in meson-meson scattering. In this case it is not at all clear from the beginning that we will end up with the 7 usual Drell-Yan result ) .

In leading order in l/Nc the following graph gives the dominant contribution. Some bremsstrahlung-type diagrams are energy suppressed, although having the same order in l/Nc .

90 R

8 c

Particle C in the final state is a heavy resonance with a kinematically de termined mass. It is stable in leading order in l/N but of course decays owing c ' to higher-order effects .

The computation of the above graph has been studied in detail in Ref . 1. Here I just give the result:

{ l) where the virtual photon momentum is

_1 o = xlS q = (q + q 1 ) - 102 /2 >'L

q 2 = xzs

and ¢ (x) is the bound state wave function determined by Eq . (2): ab 1 2 1 2 -µ--- µ - l ¢ ( = a - b ab µ 2 ¢ r + -- ¢ - � dy , (2) ab ] ab p 2 x 1 - x J (y - l 0 X) where µ is the corresponding bound state mass, µ and µ are bare quark masses . a b ¢� (x) gives the probability to find quark a with momentum fraction x. In b this way we recover with Eq . (1) the general structure as we ll as the normalization of the corresponding parton model result. This gives an explicit examp le that the parton model structure may survive the sort of final-state interaction respons ible for confinement.

The factor l/N in Eq . (1) appears as the result of a particular cancellation c ) in the hadronic four-point function6 which might not work in 1+3 dimensions . 8 ) However, for more general N /N expansions this factor has to be there in any f c case.

91 REFERENCES

1) J, Kripfganz and M.G. Schmidt, CERN Preprint TH-2266 (1977) .

2) R.L. Jaffe, Phys. Rev . D 2, 2622 (1972) .

3) G. 't Hooft, Nuclear Phys . B75, 461 (1974) .

4) C.G. Callan, N. Coote, D.J. Gross, Phys . Rev . D !1, 1649 (1976) . 5) M.B. Einhorn, Phys . Rev . D l:_!i_, 3451 (1976) .

6) R.C. Brower, J. Ellis, M.G. Schmidt and J.H. Weis, Phys . Letters 65B, 249 (1976) .

7) S.D. Drell, T.M. Yan, Phys . Rev . Letters 12_, 316 (1970) .

8) G. Veneziano , Nuclear Phys. Bll7, 519 (1976) , and contribution to this conference

92 DILEPTON PRODUCTION IN HADRON-HADRON COLLISIONS AND THE "FACTOR OF THREE" FROM COLOR

':' C. Quigg Fe rmi National Accelerator Laboratory t P. Box Batavia, Illinois 0. 500, 60510

Abstract : Recent data on inclusive dilepton production are compared with the Drell-Yan model with colored quarks.

Resume: Les resultats experimentaux recents conce rnant la production inclusive de dileptons sont compares avec les predictions du mod�le de Drell Yan contenant de � qua rks de couleur.

':' Alfred P. Sloan Foundation Fellow; also at Enrico Fe rmi Institute, Unive rsity of Chicago, Chicago, Illinois 60637. t Operated by Unive rsities Research Association Inc . unde r contract with the Energy Research and Development Administration.

93 The organizers of this session have asked me to comment upon the current state of the data on inclusive dilepton production in hadron-hadron collisions.

The question to be investigated is whether there is any evidence for or against color as a degree of freedom of the quark constituents of hadrons.

Let us recall the quark-parton model de scription of inclusive hadronic decays of a virtual photon. We assume the virtual photon to decay into a quark- antiquark pair which is dressed by the confinement mechanism, whatever that may be, into hadrons. Thus the inclusive rate is

- hadrons) - q.q:). (1) I'("yV = I'(y flavIors i V 1 1

The back-to-back quark and antiquark in the semifinal state define a jet axis fo r the final-state hadrons. Evidence for well-collimated hadron jets has been 1 obtained in studies of event structure in

+ - e e - - hadrons, y (2) V at CM emergies exceeding about GeV. Since quarks are assumed to be 4 pointlike entities, we may relate the decay rate into each quark-antiquark pair to the decay rate into muon pairs:

(3) where e is the charge of the quark flavor "i". The factor of three arises i because there are three colors of quark -- hence three distinct decay processes

-- for each flavor. Applying (1) and (3) to reaction we reconr the cele- {2) brated re sult + - cr(e e - hadrons) R 3 e 2. . + + - - 1 (4) cr(e e - ) flavors µ µ 2

94 Below charm threshold the u, d, and s quarks with charges 2/3, -1/3, and -1/3 2 can be excited, so we expect R "' 2, in good agreement with the data. Above charm threshold the c quark with charge 2 / 3 is expected to increase the total to R"'10/3 . It is not yet known how much of the new physics at 4 GeV is to be 3 attributed to charm, so we lack a definitive test of this prediction. The same manner of counting has implications for inclusive dilepton production in hadron- hadron collisions.

In the quark-parton model, the cross section for the hadronic reaction

a + b - c + anything, (5) is given by

cr(a+b-c+anything) = P (q )P (q )cr(q +q -c+anything), a 1 b 2 1 2 (6) L{q} where P . (q.) is the probability of finding quark q. in hadron i and 1 J J cr (q + q - c + anything) is the cross section for the elementary process lead- 1 2 ing to the desired final state. The summation runs over all contributing quark configurations. Drell and Yan treated the reaction illustrated in Fig. 1,

+ - a + b - + anything, (7) 1 1 in which a lepton pair of invariant mass 1 is produced with CM momentum 1/2 . • f raction x m. coll1s10ns . . at CM energy s The diffe rential cross section is given by

(8)

where the function F contains information about the quark distributions within the colliding hadrons and depends upon the scaled variable

2 T=1 /s, (9)

95 The Drell-Yan picture carries a number of significant implications. The

2 most general of these is the scaling prediction that vlf 4 da/ dvlf should be a

function of T alone. At the present time there are insufficient data outside foe

region of known re sonances (which are apparently not produced by the Drell-Yan

process) to test scaling. 5 Preliminary data of the Chicago-Princeton Collabo

ration, presented at this meeting by Shochet indicate that in pp collisions from 6 2 200 to 400 GeV/c, and at invariant masses above GeV/c , scaling holds within 5 about 20%.

The Drell-Yan model carries a second implication of expe rimental inte re st.

Consider the production of lepton pairs in TT± collisions on an isoscalar target

such as Carbon. Since the Drell-Yan process is intrinsically electromagnetic,

it impl. ies . 7 a spec ific violation of isospin invariance. Define the ratio

(10)

If the pairs are produced through noninterfering re sonances in

± TT C hadron + X ( 11)

1, but if they are produced through p

± � Tf + x c "v - L + ( 12) .I .I .

1 in gene ral. An extreme example of the latter is Drell-Yan production by p F + valence quark annihilation. For incident , the elementary process is dd � y ; TT V

for incident it is � The cross section is proportional to the square of TT uu 'ly·

the quark charge , so we find

(13) p 4

96 In a less schematic Drell-Yan model, with valence and sea quarks, the value

= /4 is attained only for rather large lepton pair masses. At very small p 1 masses, for which sea quark-sea quark annihilations are dominant, � The p 1. ratio p has now been measured, and exhibits the trend anticipated by the Drell 8 Yan model, as shown in Fig. 2. The third consequence of the Drell-Yan picture is the one which in principle tests the color hypothesis. If the quark distributions within hadrons are known, specific predictions can be made on the basis of (6) and (8) for dilepton cross sections. In the case of nucleons, data on deep-inelastic lepton scattering provide substantial information on the parton distributions, and many parametrizations have appeared in the literature. Two of these represent, in my view, reasonable extremes of parton parametrizations: the Pakvasa-Parashar-Tuan

(PPT) parametrization, with a sea quark distribution oc x 1 - x) and the 9 - (1 7 12, 10 Field-Feynman (FF) parametrization, with a sea quark distribution

1 - x) . The existence of color reduces by a factor of 3 the magnitude of oc -x (1 7 the Drell-Yan prediction because quarks of the same color and flavor must annihilate to create the virtual photon. Recent measurements of dilepton production in p-nucleus collisions at 400 GeV / c by the Columbia-Fermilab-Stony

· · . . Brook 11 an d Ch1cago-Prmceton 1 2 C olla b orat10ns d o pe rmit a test o f t h e D rell-

Yan prediction. These are compared in Fig. 3 with predictions based upon the

PPT and FF parton distributions. Roughly speaking, the data lie within the swath bounded by the two theoretical curves, a necessary but not sufficient condition for accepting the Drell-Yan formula. Even assuming that the agreement is not coincidental, the data do not select one parton distribution ove r the other, as they are confined to the small-T region in which the two predictions differ by no more than one order of magnitude. The accumulation of data at other ener-

gies and at higher values of T is of prime importance.

97 What conclusions may we draw at this time ?

(1) The Drell-Yan mechanism makes predictions which are in excellent agreement with the high-energy data. However the consequences of scale breaking in deep-inelastic lepton scattering have not yet been understood, and the nonzero mean transverse momentum of dileptons remains to be incorporated into the picture.

(2) The factor of three due to color gives good agreement with the data, but its presence is not yet proved by this process.

(3) New data are on the way in the high-mass region. The Chicago 6 Princeton and the Columbia-Fe rmilab-Stony Brook groups have in hand exten- + - sive new measurements. of and e + e - pairs, respectively. In the near µ µ future , the Columbia-Fe rmilab-Stony Brook group will measure dimuons in an 13 improved apparatus. We also await the first results on the post-J I region cjJ from the ISR.

REFERENCES 1 G. Hanson, et al. , Phys. Rev. Lett. 35, 1609 (1975). 2 J. E. Augustin, et al. , Phys. Rev. Lett. 34, 764 (1975). 3 See the talks by V. Luth, Nguyen Khanh, M. Perl, W. Bartel, V. Blobel, and

W. Wallraff at the Lep�onic Inte ractions Session for an up to date survey. 4 S. D. Drell and T. -M. Yan, Phys. Rev. Lett. �. 316 (1970); Ann. Phys. (NY)

�. 578 (1971). A relatively recent critical review is provided by J. D. Sullivan, in Particle Searches and Discoveries - 1976, edited by

R.S. Panvini (AIP, New York) , p. 142. 5 See, however, the reanalysis of data from the original BNL experiment by

L. M. Lederman and B. G. Pope, Phys. Lett. 66B , 486 (1977). 6 M. Shochet, these proceedings.

98 7 P. Mockett, et al. , Fe rmilab proposal P-332. 8 K. Anderson, et al. , "Production of Continuum Muon Pairs at 225 GeV by J.

Pions and Protons, 11 contribution to the XVIII International Conference on

High Energy Physics, Tbilisi. 9 S. Pakvasa, D. Parashar, and S. -F. Tuan, Phys. Rev. DI.Q_, 2124 (1974). 10 R. D. Field and R. P. Feynman, Caltech report CALT-68-565 (1976,

unpublished). 11 . . D. C. Hom, et al. , Phys. Rev. Lett. �. 1236 (1976); 1b 1d. 22,. 1374 (1976). 12 L. Kluberg, et al. , Phys. Rev. Lett. 22,. 1451 (1976). 13 CERN i,2. 25 (1977). CoWULieJL

p p

Fig. 1: The Drell-Yan mechanism for massive lepton-pair production in

pp collisions.

99 _

4.0 \ . - p� /J I � f *+ � - b .75 ...... - + � .5 - b PRELIMINA RY .25

0 2 3 4 5 Mµ.µ.(GeV)

Fig. 2: Ratio of µ-pair production in rr±C collisions at 22 5 Ge V / c. Data

are from Ref. 8 . The shaded band is the expectation of the Drell-

Yan model, computed over the kinematical region accepted by the

experiment. The two points away from (p,w, ¢, p', re sonances J) agree with the theoretical trend.

100 2 (GeV/c ] tn. 10 13 14 15 8 9 II 12

A-frat GeV/c 5 p+ 400 • - (Bel µ.µ. Hom,et al. · - 2 µ.}'- (Cu) {: e•e- (Be) � CP " µ:µ.- (Cu) (!) 5 N'-. E � 2 :i' 036 .,,»,., 1

fit.,, 5 t;.,,

2 7 10-3 ll',, 'i , 5 t , " ' , "

2 8 103 0 0.1 0.2 0.3 T

Fig. 3: Comparison of the Drell- Yan model with data on dilepton produc-

tion at CM rapidity y = 0 by 400 GeV / c protons on nuclear targets.

The cross sections per nucleon are displayed, assuming linear

A-dependence. Computations were performed for Beryllium; the

neutron to proton ratio of Copper is nearly identical. The solid

curve is calculated using the FF parton distributions. The dashed

curve is for the PPT parametrization. Data are from Refs. 11 and 2 12. The abscissa is T = 1 /s.

101 102 PRODUCTION OF MASSIVE DIMUONS IN PROTON-NUCLEON COLLISIONS AT FERMILAB

M. J. Shochet, Enrico Fermi Institute , The University of Chicago, Chicago , Illinois, 60637, U.S.A.

ABSTRACT 2 We have studied the production of ma ssive muon pa irs (75Mµµ ' l� GeV/c ) in proton-nucleon col lisions at Fermi lab. It was found that a very large percentage of high Pi di rect muons is produced as a member of a high ma ss muon pai r. The production cross section for muon pa i rs was also found to agree with predictions of the Drell -Yan model of quark-anti-quark annihilation.

103 The Chicago-Pri nceton col l aboration 1 has measured the production cross + section for the process p + N � µ µ- + at Fermi lab in order to determine how X often high Pi direct muons are members of high mass muon pa irs .

The apparatus is shown schematical ly in Fig. l.

I I E )1 I I I I E? I I "' I "' +> e "" u >- "" � I 0.... E � 0 "" a: I E 0 w (j) r-- _c: t- r-- +> 4- w 0 2 E 0 �I "" a: >- t- en "" u w e " ()._ I u (/) +> I "" E w e (j) ....J _c: I u 0 Vl I t- ....J ::J �I en 2 e w... I e I e I I 2 <{ I w I Ill I z I 0 t- I 0 a: I ()._ I �

104 The magnetic spectrometer2 detected particles produced near 90° in the proton-nucleon center of mass. With a hadron absorber inserted at the upstream end of the spectrometer, a detected

µ was a direct -70% of the time. In order to determine whether a di rect was µ µ a member of a high mass pair, the multihole spectrometer (MHS) was built. It consisted of ten 3. 7 m high x 1.2 m wide liquid scinti llation counters inserted in 6 m deep holes in the ground. The counters were placed along a line parallel to , but at a distance of 6 meters from the incident proton beam line. The 6 m earth shield produced an approximately uniform transverse momentum cutoff of

3 GeV/c, below which no muon produced at the target could reach a multihol e counter. The multihole detector covered a range in the center of mass of 60°

126° in polar angle and -8° - 25° in azimuth . The mass acceptance of the s apparatus was centered on tw ice the transverse momentum (P1 ) of the direct muon detected in the magnetic spectrometer. Typical mass resol ution was 2 GeV/c2 2 FWHM at M = 10 GeV/c . µµ The mul tihole counters were interrogated each time a particle was detected in the magnetic spectrometer. The time of fl ight of the multihole count wi th respect to the tri gger in the magnetic spectrometer was recorded. Figure 2 shows the results for a run with 400 GeV incident protons and with the ma gnetic spectrometer set to tri gger on particles with transverse momentum of 4.5 GeV/c.

The rel ative time of flight is plotted in units wh ich refl ect the Fermi lab beam structure - 2 nsec wi de RF bins separated by 19 nsec . RF bin number 4 was independently determi ned to be the coincidence bin. The data (Fig. 2a) shows µµ a clear signal above a flat accidental background . Figure 2b shows the time of fl ight spectrum for multihol e muons in coincidence with a pion in the spectrometer. In this case there is no signal above the accidental background .

105 0 0 0 LD 0 L!"l 0 f-· 11---,--,--·--,-1-�,--T- ,---T -, -,-- I L!"l I

c ::i iii 0 i= """ 0 c 0 01 ..0 c .,__.. ClJ tf) I u L!"l u L - u

LL a::

Cl.I I > I c � I c:D ::i I ::i I 0 0 c 01 0 -1 �· j� tf)I I I LI �--;u �.��_ _l___. __L__.i__ __,_�__,___�L j � 0 0 0 0 0 0 co <1) L!"l N

Fig. Time distri bu tions of MHS counts wi th respect to (a) a muon trigger, 2. s and (b) a pion trigger in the magnetic spectrometer (set at P1 =

GeV/c). Time intervals are pl otted in units of an RF period 4.5 of the Fermilab accelerator (18.9 nsec).

106 This 11µ data , taken simultaneously with the µµ data , shows conclusively that the

µµ s i gna 1 is real .

TABLE I s Pl Di muon Yield per µµ/µ Yield 12 )J)J 10 Protons Observed 3.75 GeV/c 7 .6 GeV/c2 55 ± 11 117 ± 20 (0.6 ± 0.1)% 4.50 8.7 152 ± 15 70 ± 7 (3.2 ± 0.3)% 5.25 9.9 34 ± 6 35 ± 6 (9.9 ± l.7)%

6.00 11.0 13 ± 4 5. 3 ±2 (12 ± 3.7)%

The raw results at 400 GeV incident beam energy are presented in Table I.

The fraction of direct muons which was accompanied by a muon in the multihole s s detector increases with increasing P1 At P1 = 5.25 GeV/c, for example, the fraction was -10%. In order to deduce the total fraction of direct muons which was produced as a member of a high mass pair, one must correct for the acceptance of the MHS . If the parent of the muon pai r had limi ted transverse momentum (�300 MeV/c) with respect to the incident proton direction, then the muon detected in the MHS woul d lie in the production plane, defined by the

incident beam line and the magnetic spectrometer. The MHS detection efficiency for the second muon wo uld then be quite high. If, on the other hand, ,the muon

pair had large transverse momentum with respect to the incident beam, then the coplanari ty condition would not be satisfied resulting in a signifi cantly lower

MHS detection efficiency. Figure 3 shows the vertical distribution of counts in

the multihole counters along with the expected distribution for limi ted dimuon

transverse momentum. From the vertical distri bution it was determined that the mean transverse momentum of the dimuon parent is " 1.2 GeV/c. In order to evaluate the detection efficiency for dimuons, a model wa s x assumed for the P1 and dependence of the dimuon production cross section: 1.6 P 4.3 E e - 1 (1 -lxl ) . �3 a dp

The P1 dependence refl ects the determined from the data . The Feynman x

107 Center Bottom Top of of of Counter Counter Counter Production + + Plane f + 3.7m _J ' r- I 50 Expected if P.i _,' en - � of Parent is 0 c � � 25 w

0 Vertical Position

00 Fig. 3. Vertical distribution of counts in the MHS . 0 ,- 10-32

.,JS 27.4 GeV This 23.7 Experiment [:0 19.4 .-- N 33 > 10 Q) (_') Pakvasa - Tuan I N E

__..(.)

-0b, �-0 r<1 -34 � 1 0 Farrar

1035 .__��-'-��---1...��___J 2 3 0. 0 . 0.4 M /.,/S

Fig. Dimuon cross sections plotted to show scal ing properties of the data . 4. The resul ts of two parton model cal culations are also shown .

109 dependence is that measured in J/� production at Fermi lab.4 (The results presented here are insensiti ve to the form of the x dependence .) Using this parametri zation , we find that, for a spectrometer transverse momentum of 5.25

GeV/c, 13% of dimuons would be detected in the MHS . Since (10 ± 2)% of the s direct muons at Pi = 5.25 GeV/c are accompanied by an MHS count, we concl ude that � v� large fraction of high P1 direct muons has high mass dimuon parentage .

The results of the experiment can be compared wi th predictions of the

Drel l -Yan model 3: quark anti-quark annihilation into a virtual photon which material izes into the muon pair. One such prediction is the scaling behavior M 3 JJJJ of the cross section M = f(·_ ) where f depends only on the µµ .QQ__dM µµ rs fraction of the center-of-mass energy carried off by the muon pa ir. The cross sections, eval uated by using the production model given above, are shown in

Fi g. The scaling behavior, i.e., data at different center-of-mass energies 4. lying on a universal curve , is observed. Results from Hom et al .4 as we ll as from two parton model calcul ations5'6 are also shown . The models differ in the assumed form of the anti -quark distribution in the nucleon . Agreement between data and the model is quite good.

The atomic number dependence of the dimuon invariant cross section has also

1•05 O.lO been measured. We ·find A ± .Q£__3 a E dp

110 REFERENCES

1 L. Kl uberg, P. A. Pirou� . R. L. Sumner, Princeton University; Antreasyan, 0. Cronin, H. Frisch, Shochet, University of Chi cago . J. W. J. M. J. 2 See Cronin et Phys . Rev. O , 181 1 (1975) . J. W. _tl. , Jl 3 S. Orell and T. Yan, Phys . Rev . Lett. 319 (1970). 0. M. �. C. Hom et Phys . Rev . Lett. 37, 1374 (1976); P.R. L., 36, 1236 (1976). 4 0. _tl. , G. Farrar, Nucl . Phys . 429 (1974) . 5 B ]]__ , 6 S. Pa kvasa, O. Parashar, and S. F. Tuan, Phys . Re v. OlQ, 2124 (1974) .

111 2 11 THE COLOUR AND FLAVOUR l/N EXPANSIONS

G. Veneziano

* ) CERN , Geneva, Switzerland

and

We i zmann Institute of Science, Rehovot, Israel

General ideas about the colour and flavour l/N expansions are presented in a non-specialized fashion according to the following layout :

Introduction 1.

A unified approach to me son dynamics 2. The basic logical scheme 2.1 Lepton-hadron interactions 2.2 2.3 Hadronic processes in lowest order Higher order effects and the Reggeon calculus 2.4

3. A possible extension to baryons

Difficulties with baryons in dual and gauge theories 3.1 3.2 A possible definit ion of dual baryons in QCD Lowest order BB and BB scattering : baryonium 3.3 3.4 Reggeon calculus for processes involving baryons

Conclusion 4.

Present address . *)

11 3 1. INTRODUCTION The aim of this talk is to explain, as plainly as possible, some recent de velopments in the theory of strong interactions which have helped the author in understanding better various concepts of hadronic physics and their mutual re lationships. I shall argue that several approaches to strong interaction physics, which have been (and are being) pursued by different groups and which seem to have little in common, are conceptually related to each other. The magic tool employed in order to achieve this unification is the use of simple, topological concepts, the claim being that , whereab for weaker interactions an expansion in powers of the coupling constant is a sensible thing to do, this is not so, in general, for strong interactions. Alternative expansions, which make use of global topological properties of graphs , can be motivated on at least three grounds. i) Aesthetic. Topological expansions use simple and intuitive mathematics and have a beautiful unifying power. ii) Phenomenological. Data, when looked at with an illuminated (biased?) eye, show dominance of simplest (planar) topologies, for example, a) exchange de generacy, ideal mixing, ZOI rule; b) dominance of short-range correlations in multiparticle production. iii) Theoretical. For mesons at least, such expansions can be formulated as l/N expansions, with N denoting the number of colours and/or flavours which 2 quarks can take. The typical expansion parameter is l/N � 1/10. The fact that quarks come in many colours and flavours provides small expansion parameters for strong interactions. Besides the already-mentioned aesthetical motivations for unification of strong interaction theories, there are practical ones. In order to clarify the potential importance of a unified approach, we have shown, in Table 1, the most apparent successes and failures of present models of strong interactions. We can easily see that different schemes are complementary rather than exclusive: each one applies to orthogonal kinematical regimes of hadron physics and usually fails where others may succeed. The conclusion is that a unified approach could cover almost all aspects of strong interaction physics. This talk consists of two maj or parts: the first one deals with mesons where, even if general ideas largely exceed actual results, a remarkably consis tent picture can be presented. The second part deals with baryons. Here, in spite of the many difficulties still ahead, some progress has been made recently.

114 Table 1

A comparison of successes and failures of present theories of strong interactions showing their comp lementarity

Theory Successes Failures

+ e e- -+ hadrons Low E, low Bj scaling PT Gauge theories Production of lepton pairs Spectrum, i.e. confinement (QCD) Large PT (?) IR divergences Point-like quark dynamics ' I'

Resonance scattering 1 "-currents Dual models Linear traj ectories Quark-like spectrum ! i QN exchange at moderate E Deep inelastic processes Exchange degeneracy, ZOI rule Large physics String models PT Vacuum exchange at high E Lack of point-like dynamics \ Superasymptopia "'-Ambiguities in definition Regge-Gribov E -+ oo, t "' physics , of input 0 models especially at critical point Proof of s-channel Scattering on nuclei (?) unitarity

2. A UNIFIED APPROACH TO MESON DYNAMICS

2.1 The basic logical scheme

This is shown in Table 2. The tool is to use l/N expansions of either gauge or dual theories. In non-Abelian gauge theories of strong interactions, out of which we shall take as the best candidate quantum chromodynamics (QCD) , one has a gauge group of colour, say SU(N ) and a global flavour group SU(N ). In nature, c ' f the colour group is believed to be an exact symmetry with N 3, the flavour group C is not known (N = 4?) and is probably broken down to SU(2) of isospin. Yet .one f can consider the limit of exact flavour syirunetry and tr at N N and the ga�ge � c ' f coupling g as free parameters in order to classify diagrams .

Following 't Hooft1 ), one shows that an expansion in l/N at g2N ,N fixed, c c f provides the conceptual link between gauge and dual theories. On the other hand , an expansion in l/N at g2N g2N fixed (and therefore N /N fixed) , provides c ' f f c the connection with Regge-Gribov theories2) . If one starts immediately from dual theories as functions of the dual coupling g and N (ther� is no room for colour D f in conventional dual models) , then the l/N expansion at g�N fixed provides the f f link between dual and Regge-Gribov models directly3) . This is just the topologi cal expansion3) [or the dual unitarizat ion4 ) ] approach pursued by several authors in recent years. We shall try to explain these ideas by examples in this lecture. I will refer the more pedantic listener (or reader) to the literature for a sys tematic treatment.

11 5 Table 2

The basic logic of unification through l/N ex pansions . The quantities in parentheses are kept fixed as the one in front goes to zero .

DRM

l/N (g2N, N /N ) QCD ��• --f=----=-��c � RFT (N colours, N flavours) c f

2.2 Lepton-hadron (meson) interactions

2.2.1 e+e- +hadrons ------We start with this much studied5 ) (-b.oth experimentally and theoretically) process . We shall consider the process in the approximations e+e- +hadrons �

� e+e- + + hadrons � e+e- + +mesons , where is a virtual photon. The y* y* y* optical theorem then relates o to the imaginary part of a + amplitude. tot y* y* The following descriptions of this process (in order of increasing comp lexity) can be considered2 l .

i) Ordinary perturbation theory (i.e. the naive quark model)

In lowest order the total cross-section is given by the imag inary part of the blob of Fig. la. Note the following features of this approximation:

= + - a) o (e e + hadrons) __.. R N Q2 , - µ- E->= c l f o (e+e- + µ+ ) f

where N is the number of colours and Q is the sum over flavours of c /:f � the square charges.

b) Quarks are produced in the final state .

- c) e+e + hadron + X cannot be described .

ii) l/Nc expansion

One takes the limit N + g2N ,N fixed . Planar graphs of the type shown C 00; c f in Fig . lb (i.e. with no extra quark loop) are selected on this limit1l. If

confinement works in the N � limit, as we shall assume , bound states of e 00

116 ---�------�--- (a) ( b)

I I (c) 2

Disc =

2=n

(d )

h h

(e) (f)

Fi g. Possible descriptions of e+e- +hadrons : a) naive quark model ; b) l/N l c expansion for CTtot ; c) l/Nc expansion for do/dPh ; d) l/N expans ion for CTtot ; e) ,f) l/N expansion for do/dPh . 117 qq are expected in the final state . They will have zero width, as explained in Section We now expect the following features to hold: 2.3.

b) No quarks in the final state . No jets either.

c) e+e- + hadron X will be given la Mueller) by the graphs of Fig. le. + (a By unitarity , the inc lusive cross-section do/dP should satisfy a con- h straint of the type : = dP (do/dP ). But one finds that the 0 tot Ih f h h r .h. s. of such an equation is of order N and not of order f I f Q� N q2 as the l.h. s. A further look reveals that , actually, do/dP c I f f h is infinite in this theory because of the zero width limit intrinsic in the l/N approximation . Hence : c d) Unitarity is grossly violated . iii) l/N expansion

The limit is now N + N /N , g2N fixed . The graphs selected in this c,f oo; f c approximation are still planar but quark loops in arbitrary number are allowed (Fig. ld) . In this approximation we expect:

b) No quarks in the final state . Probab ly jets: apart from quark loops appearing in the production amp litude (Fig . ld) , this is the same as the one usually advocated for the jets.

c) e+e- + h + X is now described by the graphs of Figs . le,f. It. is finite and satisfies the (planar) unitarity constraint with o For N /N << 1 tot" f c one finds :

N c f

since � N /N . For details see other papers2 >6) . f f c d) An approximate form of unitarity, usually called planar unitarity, is satisfied by the first term of this expansion. Some violations of uni tari ty occur, for examp le, Bose-Einstein statistics effects are neglected and J/� does not decay because the ZOI rule is exact . These effects are presumab ly small .

In conclusion, it appears that , whereas the l/N expansion is very promising c for the study of quark confinement and for the construction of the hadron spectrum, the l/N expansion is better suited for phenomenology: many of the prominent features of e+e- annihilation are expected to be a property of the leading term of the l/N expansion . - Hav ing discussed at length e+e • hadrons, we shall limit ourselves to a qu ick discussion of other important processes involving leptons and hadrons.

- • X 2.2.2 e + h e + ------

Th is is deep inelast ic electroproduct ion done, for simplicity, on a mesonic target . We shall only mention that here the l/N expansion would give a perfectly c finite and reasonable cross-sect ion with approximate (the theory is asymptotically

free N /N large enoug h) Bj scaling. The structure functions, however, will for c f be normalized to two (three) quarks per meson (baryon ) or, in other words, there will be no "sea" contribution, because this is down by N /N as comp ared to f c valence contributions. The l/N expansion will add such a term since N /N is not f c taken to be small in such an expans ion . For inclusive electroproduction, con + similar to those of e e- + h + X will hold and the l/N expansion will siderations be needed in order to have meaningful results.

For this process (inclusive high-mass lepton-pair production) the Drell-Yan 5 (DY ) mechan ism is most often advocated ) . This allows calculation of the cross

section from electroproduction data . As we vary N the Drell-Yan cross-section c (say from valence quarks) is divided by N , whe�eas the corresponding structure c function is left unchanged. If we consider the DY process in the l/N expansion, c we easily see that the relevant graphs are those of Fig. 2. They are indeed of

order l/N ' except again for a l/r factor which makes them infinite ! c

l + >_v:__

Fi g. Massive lepton pair product ion in the l/N expansion 2 c

119 If we work in the l/N expansion framework, and if, at fixed s and we M2 , make N /N we find that l/f N /N and that the cross-section is 0(1) and f c << 1, � c f not 0(1/N ), as predicted by the DY formula. This is to say that DY cannot c give the actual cross-section on top of a narrow resonance in the lepton pairs (for examp le, at the J/� mas s) which is the common belief, although it is unlike Bloom-Gilman type duality . It is still possible that, if we fix N and then we c let ,s -> 00, DY is again right, but we cannot see a reason for that to happen . M2 Good data and careful comparison wi th the DY formula will certainly teach us a lot on this interesting problem. In a contribution to this meeting Kripfganz is showing how the DY mechanism works 7) in two-dimensional QCD and will explain why it is dangerous to extrapolate this conclusion to the four-dimensional case.

The conclusion of this subsection is again that the l/N expansion could c simplify enormously the problem of confinement and lead to a first approximation hadronic spectrum. Its maj or drawbacks are : i) Gross violations of unitarity due to the zero-width limit. ii) No pionization , no jets , no sea due again to the neglect of quark loops. iii) Corrections of order N /N (i.e. quark loops) f c are neglected, in spite of the fact that this ratio is experimentally of order one .

The l/N topological expansion cures these problems without spoiling other desirable features coming from the dominance of planar graphs . On the other hand , it is expected to be very much more difficult to calculate in the l/N expansion, as exemplified by QCD in two dimensions8l . Let us now see how these ideas apply to the case of purely hadronic processes.

2.3 Hadronic processes in lowest order

If we consider first the case of the l/N expansion, it is relatively easy c to show1 l that QCD phould give, in leading order, something like the tree approxi ma tion of a planar dual model. This is because: i) planar diagrams are selected; ii) no quark loops are included . The idea that complicated planar diagrams could be at the basis of dual amp litudes is not new, of course. The new interesting point of QCD is that it gives, through the idea of confinement, a valid argument for expecting a zero-width approximation (i.e. bound-state poles in scattering amplitudes) to come out from diagrams which, to each order, nave complicated mu ltiparticle cuts (intermediate states) .

The argumen t for having no cuts in leading order in l/N is as follows 1 l . c A cut should be interpreted as a multiparticle intermediate state . However, if we stick to the planar diagrams without quark loops, perturbative intermediate states consist of an "irreducible" colour singlet state, by wh ich we mean that

120 the state cannot be written as the product of two colour singlets. Al l inter mediate states will become , after summing to all orders, single-particle states and , therefore, will have to be stable and extend in mass to infinity. This is exactly the situation in dual models, before loop corrections. Many more simi larities can be mentioned, including the one-to-one correspondence between dual loops and terms of the l/N expansion 1 ' 2) , the appearance of a Pomeron gluonium c sector2l , etc.

The l/N expansion still suffers from the problems mentioned in Section 2. 2. c One is again led to consider a new expansion2 in which N and N go to infinity l f c at the same rate. Since g2 "' g2 "' l/N this turns out to give the hadronic D c ' topological expansion of Refs. 3 and 4. It is to this expansion that we devote the rest of Section 2.

This is defined as the leading term of the topological expansion (TE) . It includes, in QCD, all diagrams of the type shown in Fig . 3a. As a result of the 3 inclusion of quark loops one has now: i) ii) do/d p is finite; iii) some r # D; sort of unitarity (Fig. 3b) is restored (e.g. partial waves are bound) . Three basic features of the tree approximation are retained, i.e.

a) ideal mixing and exact ZOI rule;

b) exchange degeneracy, hence no Pomeron ;

c) no cuts in angular momentum and no fixed poles.

Property (c) leads to short-range correlations or multiparticle production. The model for multiparticle production is a multiperipheral cluster model where clus ters are just resonances with qq quantum numbers .

Im R 0 : L n

(a) ( b)

3 a) The bare Reggeon R ; b) s-channel content of R Fig. 0 0

1 21 There is an interesting observation to make concerning hadronic mu lti plicities. Both e+e- �mesons and meson-meson scattering in the planar approxi mation produce the final state through the hadronization of a single qq pair . We shall call these one-j et processes . At this stage, we would then expect

(1)

We shall see, however, that the main high energy contribution to meson-meson - (MM) scattering comes from non-planar diagrams (the Pomeron) which modify (in a we ll defined way) Eq. (1) . Equation is still expected to hold for the mu ltiplicity (1) associated with quantum number exchange processes, which are unaffected by the bare Pomeron. Such mul tiplicity is not so easy to isolate experimentally .

2.3.3 The bare Pomeron2 •3) ------This consists, in QCD , of the diagrams shown in Fig. 4. They have a cylin drical shape where the two bases of the cylinder are made of the quark lines which come from the external particles. These diagrams introduce two related interesting effects:

i) Appearance of a leading, even signature I = traj ectory. 0 ii) Violation of the ZOI rule and breaking of exchange degeneracy for I 0 trajectories9l .

Effect (i) is welcome and leads to the identification of the new singularity with the missing bare Pomeron. Effects of the type (ii) should be small if one wants consistency with experiments.

= � Im IP0 =

(a) ( b)

Fig. 4 a) The bare Pomeron b) s-channel content of � P 0 ; o

122 Let us first discuss brief ly the properties of the bare Pomeron in the TE . These are :

It is the shadow of mu ltiperipheral, non-diffractive, unabsorbed mu ltiparticle production.

The leading J-plane singularity is a pole with intercept certainly higher 0 than the p intercept and in fact close to one 1 l .

The s-channe l state is a "two-j et" state (Fig. 4b) . Assuming equipartition of the energy between the two weakly correlated jets, one finds:

(2)

Such a relation , first suggested in the framework of Preparata's massive quark model 11) , seems to be in agreement wi th the availab le data, if one takes I\r, from non-diffractive inelastic pp cross-sections and allows for one half of the energy to go on the average to the leading protons .

The slope of the Pomeron trajectory (i .e. shrinkage) is small in spite of the high mu ltiplicity. This seems to be against naive random walk arguments, giving higher slopes for higher multiplicities . It is again a consequence of the two-jet structure of the process.

The s-channel process reminds us also of the Low-Nussinov mechanism12l for the bare Pomeron . Through the exchange of gluons the incoming quarks swap their colour : this leads from the original two low-mass colour singlets to two high-mass [o

Finally the question of the internal quantum numbers of the bare Pomeron, which is related to the breaking of the ZOI rule . The graphs of Fig. 4 introduce mixing among the various quark states of I= O ; (uu + dd) , ss , cc , etc. The bare Pomeron itself is a mixture of such states. The mixing angle (s) turns out to depend strongly on t and can be computed in explicit models . One finds 1 3 l that , at t = 0, is about midway between a pure �o SU (3) singlet and an ideally mixed f, but has almost no charm content. Hence S -1 2 oTi oK o · At t GeV becomes very close to an singlet p 2 p >> �p �o SU (3) giving

do " do » do dt T dt dt l ip l Kp l �p

Finally at t GeV2 becomes ideally mixed (i.e. pure u and d quark) . 2 1 �o This is what makes f1 + TITI/f1 + << 1 and saves the ZOI rule. KK

123 The lesson to be learned is that ZOI violations and the Pomeron are related . A good model should make the first small without killing the Pomeron . The non-planar effects giving rise to both effects are strongly t-dependent . This is what Chew and Rosenzweig call asymptotic planarity1 3 ) and what field theorists relate to asymp totic freedom5 l . There is an outstanding question. How do these considerations apply to unnatural parity states? Why is the ZOI rule much worse for them? Can the U(l) problem be solved at a finite order in l/N? For the moment there are only partial, unsatisfactory answers to these important questions .

2.4 Higher order effects and the Reggeon calculus

In Section we have defined the bare Reggeons and the bare Pomeron as 2.3 sets of leading diagrams in the l/N expansion. The interesting point about this definition is that the higher order corrections can be shown to give the Reggeon calculus of Gribov 1 4 ) in a way which is manifestly consistent with t- and s-channel unitarity.

As an examp le, consider the set of diagrams depicted in Fig. 5. It gives the lowest order contribution to the two-Pomeron cut and is of order l/N2 rela tive to the bare Pomeron itself . We say that this contribution has b = 2 and h = 1, meaning that the external lines couple to two quark loops and that the least complicated surface on which we can embed the graph in a sphere with one handle (i.e. the torus) . In this terminology the bare quantities are those with 0. 2, h = Our term with b = h = 1 gives, in terms of s-channel processes:

i) Elastic and diffractive dissociation cross-sections. Notice that o / O(l/N2 ), and , in a model calculation of Bishari1 5 l , indeed e1 otot MM oMM;o l/N2 , where , for scattering, N N = 2.6. el tot � TITI = eff ii) Absorptive corrections to ordinary non-diffractive events. iii) Events with double multiplicity in the central region and also events with unassociated production of � or J/� particles .

The systemati cs1 6 ) of this expansion in l/N2 leads to a "derivation" of Reggeon field theory from field theory . The bare Pomeron intercept a turns out to p o differ from one by a quantity of order one (but small for dynamical reasons), whereas the triple Pomeron coupling is of order l/N and the n-Pomeron coupling n 2 of order (l/N) - • As a consequence, we do not expect to be at the critical point of Reggeon field theory a (O) = l and for large N, one is pushed to [ ren ] the supercritical situation recently investigated by several authors1 7 l ,

Not only is s-channel unitarity explicit . One can even introduce a cut Reggeon field theory1 8 l , in which properties of the s-channel intermed iate states can easily be computed and consistency with unitarity under various approximations can be checked. 124 Fig. 5 Lowest order contribution to the two-Pomeron cut

As a final remark we mention that the process of double Pomeron exchange

comes out in this scheme ·at the h = 2 level [i.e. O(l/N4)] together with the three-Pomeron cut . The situation , however, is no t that simple for processes in volving baryons (see Section 3.4) . Surnn1arizing again the TE for mesons2 •3•16l , we can say that any S-matrix element involving n external mesons can be written

as n b,h b 2h S s (l/N) + n L L ' b=1 h=o

where

b gives the flavour structure of the contribution [e.g. the ZOI rule is violated (b - 1) times],

h gives the J-plane structure [for a (p + 1) Pomeron cut one needs h p] . � This is related to long-range correlation effects in multiparticle production.

Both parameters b and h have therefore an obvious physical meaning and hadron phenomenology seems to suggest the decreasing importance of terms with higher b and h, at least in most of the accessible kinematical regions.

3. A POSSIBLE EXTENSION TO BARYONS

3.1 Difficulties with baryons in dual and gauge theories

Difficulties with baryons in dual theories go back to the early days of duality with the paradox of Rosner1 9 ) , which is best illustrated in the language of duality diagrams . Clearly , in BB scattering , ordinary qq states in the t-channel have to be dual to states with qqqq quantum numbers in the s-channel. If these have to be interpreted as single-particle states , then we are faced with the need for exotics. On the other hand , one has the examp le of the bare Pomeron of Section 2.3, in which the qqqq s-channe l quantum numbers are carried by a continuum (or two resonances) with no need for exotics . What is the situation in BB scat tering? Recently this problem has received renewed attention following two very 0 stimulating papers of Rosenzweig and of Chew2 ) , who introduced the concept of narrow "baryonium" states in analogy with "charmonium" . Other papers dealing with baryons, both in the framework of the TE2 1) and in that of the dual string picture22 ) have contributed towards an improved understanding of baryon dynamics.

In this lecture we shall follow an approach that tries again to tie up dual and gauge theories and wh ich is due to Rossi and the author2 3). The problem is that the l/N gauge theory approach cannot1 ) be trivially extended from the c mesonic case to the baryonic one . This because, in SU(N )' the baryon is made of c N quarks and, as a consequence, the gauge-invariant sets of diagrams to be con c sidered are different for different values of N and cannot be compared with each c other1 •23) . In particular , a l/N expansion is not feasible . An alternative is c to generalize directly one of the main consequences of the l/N expansion for c mesons (rather than the expansion itself) , i.e. the zero-width approximation wh ich was related (see Section 2.3) to the appearance of "irreducible singlets" in all possible channels. This is the approach2 3 ) that we are going to follow.

3.2 A possible definition of dual baryons in QCD

We shall define dual QCD baryons in analogy to dual QCD mesons of the qq and quarkless (gluonium) type. This is shown in Table 3. For each type of hadron one can associate a certain type of "irreducible" (local) colour singlet operator built from the minimum number of quark fields q(x ), q(x ) and from an indefinite i j number of gluon fields A (y) , where y lies on certain paths . One can associate µ with each operator of this kind a string picture where the string is identified with the path. We thus obtain the usual open and closed string picture24) for qq mesons (called here M ) and gluonium or Pomeron-like particles (called M0). 2 At this point the generalization to ordinary baryons (called B3 ) is straightforward and is given again in Table 3. Notice that a Y-shaped baryon comes out and not , say, a 6-shaped one . The string "junction" is interpreted in terms of the E tensor E " This is the new element when baryons are considered (notice that ijk for arb itrary N one has a star-like configuration for B with an N -ple junction c 3 c corresponding to an E-tensor with N indices) . Dual QCD baryons will then be de c fined in terms of a set of graphs such that all intermediate states are of the

type*) E ..• Aq) •.. Aq) ... Aq) l O ). From this definition one ijk (AA i (AA j (AA k can proceed2 3 ) to the construction of scattering amplitudes involving baryons.

) *) This idea is not dissimilar to that of gluon chains, see Tiktopoulos2 5 .

126 Table 3

Ordinary mesons and baryons: colour structure and string picture

Hadron Gauge invariant operator String picture

= µ M2 qq meson q(x ) exp g dx A q (x2) x, X:z. 1 µ � ) ¥ 'l ( rXl

= µ Mo quarkless meson Tr exp g dx A ( f µ ) 0

Eiik[•xp (g]1 dVµ) qcx1i]x B, = qqq baryon x ' � )( x [•xp (gr dvµJ q(x,)l [·xp ( g T dvµJ q(x,)J x J x k iy� ){,� ·z

3.3 Lowest order BB and BB scattering: baryonium, BB annihilation and a new ZOI rule

Since we lack a complete topological . classification of QCD diagrams with baryons similar to the one for mesons, we resort to a dual-inspired technique in order to construct scattering amplitudes . For instance, we obtain BB scat tering through the following steps :

i) Define the coupling of ordinary currents to BB, which is conceptually trivial.

ii) Obtain the coupling of mesons to baryons from factorization in the current channels. iii) Construct BB and BB scattering amplitudes by sewing up BBM vertices. This is shown pictorially in Fig. 6.

L M M =

Fi g. 6 Construction of the BB scattering amp litude via meson exchange

127 At this point we can ask what are the s-channel intermediate states in Fig. 6 which are dual to ordinary M mesons . We find that they all have the colour 2 structure of the states

and not that of a two M2 continuum (i.e. as a product of two colour singlets) . It is in order to indicate this fact that we have drawn the extra dotted lines in Fig. 6. In the string picture this dotted line can be thought to represent the motion of the string junction . The fact that the oper,1tor in Eq. (3) is again an "irreducible" singlet leads us to associate these states with a new family of zero-width mesons which we denote by M� (J for junction, 4 for the total number of quarks) and call "baryonium" states20l .

We conclude that the old Rosner problem1 9 ) cannot be solved by identifying �xotic intermediate states wi th a continuum. We emphas ize that study of the colour structure of the diagrams has been crucial in order to reach this con clusion . More generally, we can say that , whereas ordinary duality diagrams show just the flow of flavour , for a comp lete description of baryonic processes, further information is needed precisely on the flow of certain colour indices. This leads to a "doubling" of duality diagrams in BB and BB scattering . The complete set is shown in Fig. 7, where the labels S and A stand for scattering and annihilation in the s-channel, respectively. This is because, once quark loops are turned on J in the spirit of the TE , M (baryonium) states yield BB + mesons as intermediate states. We see that scattering and annihilation channels are dual to each other in BB scattering. This means that ordinary Reggeon and Pomeron exchanges control scattering processes, as expected. It also means , however , that high-energy annihilation is controlled by the exchange of baryonium traj ectories.

Consistency requires three new fami lies of baryonium states shown in Table 4 (in analogy with Table 3) . One can have an estimate of the intercepts and slopes * of their trajectory and the result ) is shown in Fig. 8. The crucial question here is: do these states stay narrow or not after unitarity corrections are in cluded? If we limit ourselves to the planar quark loop these states obey a new ZOI rule by which the two string junctions they possess cannot annihilate each other and therefore they will decay into B + B + mesons. If close to (or below) the BB threshold, these states will be narrow (or actual bound states) in this lowest order approximation . Before discussing corrections, however, we have to see how BB annihilation proceeds in our scheme .

*) Our values are not incompatible with those obtained mar� recently by Dosch and Schmidt2 6 l .

128 . )�-----_�[ ] � . J (S, ) c

__ ...

Jj (A,)

' \ /'' \ I I I I I II ,, \\. --

Fig. 7 Components of BB and BB scattering in lowest order

i 29 Table 4

The three baryonium families: colour structure and string picture . The exp symbol represents the same exponential as used in Table Cf�) 3.

Hadron Gauge invariant operator String picture

' x M� = baryonium with x.• xp p qqqq QN EijkEi 'j 'k' q e . ex . [ [JJ] [q (JJ] )( .:I /E E i J X.i. 'l 'l- . [.,,rt( [··· r11:r [··' {J �;. J:i.

x K EijkEi' j 'k' [ exp = q baryonium with (f M� y J] i qq QN

x x� •c:>< .'} €: - exp exp exp ci i [JX J( lf)X kr X1 JqJ' [ j [ [ [f

I x EijkEi j 'k' M� = quarkless baryonium · [···rlf[·· ,lf( [ ..,r J( ·Gf t'

2 3 4 5 ( 1/cx' units ) M2

Fig. 8 Baryonium traj ectories before mixing

I 30 Except for a rather uninteresting annihilation into gluonium, BB annihilation consists of three incoherent processes2 1l which we can call one-j et- , two-j et-, and three-j et-annihilation . At high energy the three processes are controlled by exchange of M� , M� , and M� baryonium and are therefore in order of increasing magnitude (see the intercepts in Fig. 8) . We also expect the multiplicities to be in well-defined ratios . Assuming equipartition of energy in the jets we have

- n <9s) 3 n (4s " 3il + (s) 3 jets 2 jets e e - (4) � 2 ) �

The comparison of the first and last terms of Eq. (4) , assuming high-energy annihilation to be three-j et dominated , appears2 3 ) to be in agreement with the data. Another interesting point : we find2 3l, from a study of the impact para meter structure of annihilation , that the three-j et process is the most central

(low J) and the one-jet the most peripheral.

This last remark allows us to go back to the discussion of baryonium decay. If annihilation is dominated , even at low energies, by the three-jet process, as already speculated many years ago2 7 ) , then it is confined to rather low partial waves. Since there are M� states of spin up to three or four near the BB threshold we expect these to be little influenced by annihilation and to remain narrow and quite elastic . States of this family could be perhaps identified with the narrow quasi-elastic bump s recently found in low-energy pp experiments2 8l . J I = 2 The crucial test of this idea would be , of course, to look for an M4 meson of mass close (because of approximate exchange degeneracy) to that of the recently

found bumps 1940 MeV) for example in the reactions pn + TI+ -- or (� + x - + TI n p + X--

3.4 Reggeon calculus for processes involving baryons

Leaving details out (see Ref . 23) one can just say that , within the present scheme , some justification can be found for the quark counting rule2 9 ) in total cross-sections. The result can be summarized by saying that the coupling of the bare Pomeron to a hadron is given by

9P q qI h o iqi iE where c are of order one but are different for non strange, strange , and charmed i - quarks . Since for a baryon the includes N terms , we see that the Pomeron lqiEh c coupling to baryons is of 0(1) and not of 0(1/N) like the Pomeron coupling to mesons or the triple Pomeron coupling . As a consequence , l/N-type expansions do not give, for processes involving baryons, the straightforward Gribov calculus perturbation theory1 4) . According to the process and to the kinematical regime in log s considered , various types of expansions emerge3 0l, some of wh ich are of the type recently considered for scattering on nuclei.

4. CONCLUSION

A short conclusion for a long talk. Using topological properties of Feynman and dual graphs, a considerable unification of hadronic and parton concepts can be achieved in lepton-hadron and hadron-hadron interactions in all their mo st interesting aspects.

Much more work is certainly needed, especially concerning baryons, and every effort should be made to put the consequences of the scheme under stringent ex perimental tests. The logical beauty of the approach , which unifies the most successful tools now available for hadron physics , and the basically sound phenomenological picture emerging seem to indicate that a joint effort in this program will not fail to produce further rewarding results in the near future .

I wish to thank Professor Tran Thanh Van for having provided such a warm and stimulating atmosphere at the Rencontre in Flaine .

132 REFERENCES

1) G. 't Hooft, Nuclear Phys . B72, 461 (1974) .

2) G. Veneziano, Nuclear Phys . Bll7, 519 (1976) .

3) G. Veneziano , Phys . Letters 52B, 220 (1974) ; Nuclear Phys. B74, 365 (1974) .

4) H.M. Chan , J.E. Paton and T.S. Tsun, Nuclear Phys . B86 , 479 (1975) . H.M. Chan , J.E. Paton, T.S. Tsun and S.W. Ng, Nuclea;:-Phys. B92 , 13 (1975) .

5) See, for example, J. Ellis , Deep hadronic structure, lectures given at Les Houches Summer School , 1976.

6) M.B. Einhorn, Univ . of Michigan preprint UM-HE 76-36 (1976) .

7) J. Kripfganz and M.G. Schmidt , CERN preprint TH 2266 (1977) .

8) G. 't Hooft, Nuclear Phys . B75, 461 (1974) . C.G. Callan, N. Coote and D-:-Y:- Gross, Phys. Rev. Dl3, 1649 (1976) . M.B. Einhorn , Phys. Rev . Dl4, 3451 (1976) . R.C. Brower, J. Ellis, M.c;:-S chmidt and J.H. Weis, Phys . Letters 65B, 249 (1976) : CERN preprint TH 2283 (1977) . M.B. Einhorn , S. Nussinov and E. Rabinovici, Univ . of Michigan preprint UM-HE 76-39 (1976) .

9) G.F. Chew and C. Rosenzweig, Phys . Letters 58B, 93 (1975) ; Phys . Rev . D _!2, 3907 (1975) . C. Schmid and C. Sorensen, Nuclear Phys . B96 , 209 (1975) .

10) H. Lee, Phys . Rev. Letters 30, 719 (1973) . G. Veneziano , Phys . Letters 43B, 413 (1973) .

11) N.S. Craigie and G. Preparata, Nuclear Phys . Bl02, 497 (1976) .

12) F.E. Low, Phys . Rev . D 12, 163 (1975) and these proceedings.

S. Nussinov, Phys. Rev .----"Letters �34, 1286 (1975) ; and Princeton IAS preprint C00-2220-57 (1975) .

13) G.F. Chew and C. Rosenzweig, Nuclear Phys . Bl04, 290 (1976) . M. Bishari , Phys . Letters 59B, 461 (1975) .

14) V.N. Gribov, Soviet Phys . JETP 26, 414 (1968) . For a recent review see, for examp le, M. Moshe, Univ. California San Diego preprint IOPI0-166 (1976) .

15) M. Bishari, Phys. Letters 66B, 54 (1977) .

16) G. Marchesini and G. Veneziano , Phys. Letters 56B, 271 (1975) . M. Ciafaloni, G. Marchesini and G. Veneziano , Nuclear Phys. B98, 472 (1975) .

17) See, for example , D. Amati, M. Le Bellac, G. Marchesini and M. Ciafaloni, Nuclear Phys . Bll2, 107 (1976) . A.R. White, CERN preprint TH 2259 (1977) .

18) M. ·Ciafaloni , G. Marchesini and G. Veneziano , Nuclear Phys. B98, 493 (1975) . P. Suranyi, Phys. Rev. D _!2, 1064 (1976) . 19) J.L. Rosner, Phys . Rev. Letters 3.!_, 950 (1968) .

1 33 20) Rosenzwe ig, Phys. Rev. Le tters 36 , 697 (1976) . C. Chew , LBL preprint 5391 (1976).- G.F.

Eylon and H. Harari, '.'iuclear Phys . 880 , 349 (1974) . Y. 21) ll .R. Webber, Phys . Letters 62ll, 449 Nuclear Phys . B117, 445 (1976) . H.M. Chan and T.S. Tsun, Ruther ford La(Im)borat; ory preprint RL-7(1-054 (1976) .

22) Artru , Nuclear Phys . 885 , 442 (1975) . X. M. lmachi, S. andf. Toyada, Progr . Theor. Phys. 54, 280 (1975) ; 551, 1211Otsuki (1976) ; 517 (1977) . 22, l!_,

23) G.C. Rossi and G. Veneziano , CERN l'H 2287 , to appear Nuclear Phys ics B. preprint in

24) For a review, see Rebb i, Physics Reports _1_� , (1974) . C: . 1

25) G. Tiktopoulos , Phys . Le tters 66B, 271 (1977) .

26) H.G. Dosch and Schmid t, CERN preprint Tl! 2296 (1977) . �LG.

27) ll .R. Rubinstein and H. Stern , Phys . Le tters 21, 447 (1%6) .

28) CERN 74- 18 (1974) , p. 46 . A. Astbury , V. Chaloupka et al ., Phys. Letters .§l_ll, 487 (1976) . Br�ckner et al ., Observation of a resonance near pp tl1reshold (CERN W. preprint , 1976) . narrow

29) F.M. Levin and L.L. Frankfurt , JETP Le tters 2, 65 (1%5) . H.J. Lipkin and F. Scheck, Phys . Rev . Letter-;;- �, (1966) . 71

J. Cardy, Nuclear Phys . B75, 413 (1974) . JO) M.S. Dubovikov and K.A. Mar tirosyan , CfRN preprint 2262 (197f>) . TH Schwimme r, Nuclear Phys1�. B94, 445 (1975) . A.

I J4 COMMENTS ON THE PCMER01'

F. E. LOW

LABORATORY FOR NUCLEAR SCIENCE AND DEPARTMENT OF PhYSlCS �1.1 ssachusc�tts Institu te of Te chnology Cambr idg e. Massnchusvtls 02 IJ9

AllSTRACT

in my talk today I wi sh to discu ss three related subj ects. The first is m d l pomeron wh ich su z8ested sever;:1 ! a o e of Lile: I year s .J go (and wh ich was also su gg12ste

135 I. The pomeron model

(a) This consists in first aooroximation of the exchange

of a single colored vector gluon between high energv particles

made up of quarks in color singlet states . The original mo tiva-

tion was very simple . First: vector (spin 1) exchange leads

to a constant cross-section if one integrates the differential

cross-section given by fig . 1. Of course , this cross-section is

purely elastic . More on this later .

g2 = grl� . g

Fig. I

Second : the real part of the p-p amplitude is small - hence a C) conventional vector exchange (odd will not do , and one must

IT have a forbidden quantum number (think of exchange between

IT deuterons ; the single exchange between deuterons vanishes ).

Hence the beauty of color . Thus , the mode l builds in constant

O (log's etc . are higher order corrections) and zero real part . T The fact that these are not too bad may be a hint that higher

orders are not as important as one might have guessed

1 36 g 2 1 Be fore I mention the successes of the 4iT 2 mode l, let me give its main problem : the re is no colored gluon theory , hence , for any real calculation , severe approximations are needed. The ones I have used are more or less based on the

MIT bag mode l, although presumably any initial hadron wave- functions could be used . However, the MIT bag has been very successful in low energy phenomenology ; indeed one hopes the bag may replace the very strongly coupled infra-red region of color exchange , and hence validate a perturbative approach . However , this is a semi-classical model, and hence very difficult to make quantum corrections .

(b) In more detail : Figure 1 is replaced by Figure the 2, amp litude so calculated is squared and summed over the excited states of the bag . This is the total excitation cross- section at impact parameter b. The decay proceeds differently , and will be described presently. Note that we do not necessarily need to know the decay processes to calculate the elastic cross-section - think of decay 6 years after the formation of the parent a 10 nucleus . excited octet states of each bag ( n) 2

quarks / (9 exchanges)

o-(b)=I - n b-

f f hadron hadron

Fig. 2

137 Briefly, qualitatively, we have two octets racing away

from each other, but linked by octet flux lines , as shown in

Fig. 3 .

- t = r

transverse momentum gluon exchange t = 0

line

t = T

Fig. 3

In the next part of our talk we will describe the decay of such a system in a very simple (but covariant and quantum) model.

= i) 0T constant (input)

ii) = 0 Re f (input )

1 38 (iii) Approximate s channel he licity conservation - non-

tr ivial , and an explicit calculation . It depends on the non-

� trivial identity µ + µ = e/2m . Otherwise , no extra para- p N N meter .

iv ) Angular distribution , geometrical , and one finds a

natural derivation of the Chou-Yang eikonal function - give s lOt � e , again , no extra parameters.

v) Based on the simple decay mode l previously mentioned ,

uniform rapidity distr ibution in the central region , and hence

logarithmic multiplicity .

( vi) t = diffraction disociation is not forbidden . 0

II (a) How does the system described above decay? How long does

it take ?

Consider instead of quarks (or racing apart , linked by 6 5 ) octet flux lines, the simp ler system of qq racing apart linked

by an electric flux line - and further , consider only the one

dimensional system . We thereby have automatic confinement ,

since the potential energy increases linearly with separation .

How does the system break? Presumably by tunneling : quarks

tunne l through the forbidden region la Klein paradox, but the a calculation in these terms is horrendous - the walls are moving ,

the tunneled quarks screen the wa lls , etc . Fortunate ly , for

m = O, the system is exactly soluble , as shown by Schwinger . q The spectrum cons ists of a single neutral particle , the meson ,

139 2 of mass m There is no interaction . Nevertheless , we a/� . can create a qq state as shown in Fig. and ask for its de cay 4 spectrwu distribution , and asymptotic position of the produced

particles .

-p p • q q

Fig. 4

All these can be calculated exactly - the last of course requires a wave packet.

(b) Answers are the following :

i) rapidity dis tribution . As long as the rapid ities are a fraction � y' when is the max imum rapidity and � 1, i Y i <

+ 00 Of then as Y all distributions are independent of the y's. course this tota l absence of correlations is a consequence of the lack of interaction .

( ii) number distribution . If we let the norm of our qq state be independent of energy , then the number distributions look rather mu ltiperipheral:

with zc = 1, so that ZP = 1 as we ll . i 1 (iii) Spatial location of particles: each meson of ve locity v is located at x = v t, just as if every meson had been made i i i at t and x 0. 0 iv) We can now ask: whe re did each meson sp ring loose from the expand ing fire-sausage (fire-condensor )? Me sons of successive rapidity , say y and y , will cease to interact (here 1 2

i40 we imagine the theory to be complicated by a short range -

say one fermi-interaction) when, in the ir relative rest system,

the ir separation has grown to one fermi ! If we call the cor-

responding time T, we find that the time the slower particle ,

y , detache s is t a very long time . 2 With y - y 1, t cosh y 1 2 � � (c) Comment on general production amp litude : if it has

no sharp energy dependence compared to a suitable wave-packet,

then x = v t is a general property of the asymptotic wave i i function, and the above description holds . One has only to add

uniformity in y to obtain the picture of times and distances

� cosh y. This is incidentally the Gottfried space-time picture .

It is of course possible for a scattering amplitude to have un-

usual energy dependances - e.g. a resonant phase , so that a

large time delay 6t 35/aE over the above time s will occur , �

e.g. x = v(t - J6/3E) - where for example ao/aE can be millions

of years; again a decay is instructive . In particle collisions ,

we have virtually no phase information , so that in princ iple

one might have such delays ; there are , however , no corresponding

long range correlations in the cros s-sections .

III. Can one measure the se sizes and times?

In principle , this should be pos sible - we know for example

of Gottfried 's discussion of nuclear processes. In addition ,

consider the green 's function G(x,y) =< in j(x) j (y) in> l I connected '

141 where x and y are far apart . Then in principle ik · x G(k) =cf'e G(x/2 , - x/2) dx should be rapidly varying over wave-numbers k "' f-- In fact, G(k) is exactly the max bremsstrahlung amplitude for the state l in> , so that one might -l hope to see a variation from the dk/k spectrum at k fm x l/cosh y, � l as we ll as a pos sible Hanbury-Brown-Twiss effect in the double-

bremsstrahlung .

For a variety of technical re asons , connected with charge

conservation , the se effects are very hard to find in the theory ,

and may very we ll conspire to conceal themselves, unle ss anomalous

phases are present .

142 JET PRODUCTION AND THE DYNAMICAL ROLE OF COLOR*

Stanley J. Brodsky Stanford Linear Accelerator Center Stanford University, Stanford, California 94305

Abstract: We discuss how multiparticle jets which originate from quarks, multiquarks, hadrons , or gluons can be distinguished by their (a) quantum num ber retention, (b) leading particle distributions , and (c) hadronic multiplicity. The possible quantitative connection between initial color separation and hadron multiplicity is emphasized. Evidence for wee quark exchange as a dominant hadronic mechanism is presented. We also discuss several new mechanisms for gluon jet productioll.. Finally, we consider the possibility of utilizing a high multiplicity trigger to expose gluon exchange and production contributions in hadron- and lepton-induced reactions.

Resume: Nous discutons comment les jets de multiparticules dont l 'origine est SOit des quarks , des multiquarkS des hadrons OU des gluons peuvent etre distingues par (a) leur retention de nombres quantiques leurs distributions de particules dominantes (c) leur multiplicite hadronique. (b)Nous mettons l 'accent sur la liaison quantitative possible entre la separation des couleurs initiales et la multiplicite hadronique . L'evidence pour l'echange de wee quarks comme un mecanisme hadronique dominant est presentee. Nous discutons aussi plusieurs nouveaux mecanismes pour la production de jet engendre par un gluon. Finale ment, nous considerons la possibilite d'utiliser la detection d1evenements de haute rnultiplicite pour mettre en evidence les contributions des echange et production de gluons dans les reactions induites par des hadrons et des leptons.

*Work supported by the Energy Research and Development Administration.

143 I. Introduction

The production of multiparticle jets appears to be a common feature of all high en 1 ergy hadron- and lepton-induced reactions. Despite the indications that their average 2 multiplicity and transverse momentum distributions are surprisingly similar, the underlying quark and gluon content of these jets can be quite diverse. In particular, we - expect quark fragmentation jets of various flavors in e e -hadrons , as well as in the +

current fragmentation region of lepton-induced reactions. We also expect di -quark (qq) and multiquark jets in the target fragmentation region in deep inelastic scattering as well as in the beam and target region in Drell-Yan massive pair production. In the case of large P reactions, one expects quark, multiquark, and even jets of hadronic parentage T depending on the hard scattering subprocess. Even more intriguing , if one takes quantum chromodynamics at face value , one must expect at some level, the production of jets corresponding to gluon fragmentation in any of these processes. In the case of ordinary forward hadronic reactions, the jet-like forward and backward multiparticle systems are usually considered to have a conventional hadronic origin, but as we shall discuss in

Sections VI and VII, the primary parents of these systems could well be colored, if the initial hadronic interaction can be identified as being due to gluon or quark exchange.

One of the important phenomenological questions in particle physics, then, is how to empirically discriminate between jets of different origin, i.e. , how to distinguish the different flavor, color, number of quarks or gluons of their parent systems. this In paper I will discuss a number of discriminants , including dn/dy (the height of the rapidity plateau in the central region) (Section VII), the fragmentation properties (the power-law falloff and quantum numbers of leading particles at x- 1) (Section IV) , and the possible retention of charge and other quantum numbers (Section II) . The multiquark jet which is left behind in the target fragmentation region in deep inelastic scattering and theDrell

Yan process is especially interesting because of theoretical uncertainties regarding the composition of the hadron's parton wavefunction. We discuss special tests for such sys tems in Section II. Finally, in Section VII we speculate on thepos sibility that the initial color separation controls the height of the multiplicity plateau , and that events with large multiplicities are sensitive to gluon exchange contributions.

144 3 Charge Retention by Quark and Multiguark Sys tems II. 4 Feynman originally proposed the elegant ansatz that the total charge of a jet

(2. 1)

could reflect, in the mean, the charge of its parent. However, as noted by Farrar and Rosner, 5 this connection can fail in specific model calculations, and accordingly there has been little subsequent interest in using this method as a jet discriminant.

In order to see why exact charge reten-

M tion fails, consider the simple model for

+ - e e - -qq - hadrons shown in Fig. 1. The 'Y 1 rapidity interval lyl y �- log s is filled < max 2 uniformly by the production of neutral gluons , which subsequently decay to qq pairs. These Fig. 1. Simplified model for the rapidity distribution of virtual gluons and then recombine with the leading quarks to mesons in e+e- -qq - hadrons. See also Section III. produce mesons. we could cut through a If meson and sum all the charges to the right of the point Y ' then clearly Q (Y >Y )=Q . a J a q However, since we must sum over hadron charges an extra quark is always included in 3 the sum, and the total charge corresponding to the antiquark jet is • 5

(2. 2)

1 where 1J = is the mean charge of quarks in the sea: =-(2/3 - 1/· 3) = 1/6 for an Q q sea 1JQ 2 SU(2)- (or SU(4)-) symmetric sea, and = (2/3 - 1/3 - 1/3) 0 for an SU(3)-symmetric 1J Q � = sea. Actually this result is quite model-independent, and Eq. (2. 2) will apply to all jets which need a quark for neutralization. More generally, for any conserved charge A=Iz , 3 B, S, etc. , one predicts • 5 (2. 3) where is the quantum number of the parent quark or multiquark system and the sign is A J + ( ) the parent needs a quark (antiquark) to neutralize it. Notice that is independ - if 1JA 3 ent of the process and of the jet type and is a universal number. The result (2. 3) is un

affected by resonance decay or baryon production. From parametrizations6 of the quark

145 u MlllVJN.NV

(xbi � 0.2) d

= p Q 4/3 { .'!.+773 Q Fig. Parton model diagram 2. for vp-µ-x as viewed 1---- in the w+p c. m. sys �logs� tem for the valence Fig. 3 . Idealized distribution for w+p - X -µ-X) (z;p region (x ;: 0. as s - oo. bj 2). distribution functions, one can already

+ • h + h > 5 determine empirically that ?J "" . 07, cor l:P, Q O o h + - h- ¢ > 90° responding to partial suppression of the � protons s > 25

30 strange and heavy quarks in the sea. For 101 t ci t 1 the usual quark models , ?J =O. � Ht 1JB=3" I en z c 20 t t Given the fact that the ?J are uni-

For example, Fig. shows the expected 0.4 2 + 0.3 + initial quark flow in the W p c.m. sys- ct"" - 0.2 tern for vp -µ x, in the valence quark t 0. 1 region and Fig. 3 indicates the -4 -3 -2 -I 0 2 3 4 �j � O. 2, Yc.m. predicted hadronic charge distribution Fig. 4. Data from the Fermilab 15 ft bubble + - dN /dy - dN /dy expected at very large chamber for z;p -µ+x. 7 The rapid ity distribution and the charge struc 2 - s = (q+p) . For x <0. the presence of ture d�/dy - dN /dy of the final bj 2, state are shown, as well as the + sea quarks which can be hit by the w transverse momentum distribution of the emitted hadrons . The curve tends to further increase the asymmetry for

is from pp reactions . T between the two hemispheres . Taking

?J =0.07 , this implies that the plateau height will be times larger in the target (uu) Q �2. 3 7 fragmentation region compared to the current (u) fragmentation region. The data (Fig. 4a)

146 Tr 10 c - 1/3 u "'O K s ----Vh�----'-- } = - o+ Q 1/3 Q c 1;3 l d p l u

(a) lb) Fig. 5, The Drell-Yan mechanism and the Fig. 6 . The Drell-Yan mechanism for expected charge distribution for K p - £+r + X in the valence n±p in the valence valence- region (x ' x - £+r + X K p valence region (x , x ?: 0. 2) . rr p ?: 0. 2) . seem consistent with this prediction. It is also interesting to note that the transverse + momentum distribution of W -induced events is the same as that measured in pp co!lisions, shown by the solid lines in Fig. 4b .

One of the most interesting areas of application of the charge retention technique is to confirm the underlying quark structure predicted by the Drell-Yan process. 8 For example consider in the valence quark reg'ion (x , ?:, x - x ' rr±p -t£-X a � 0. 2; a � � L x 2 /s). The predicted quark-flow diagrams and the charge distribution of the a� � .At final state hadrons are shown in Fig. 5 . We estimate that the width of the charged fragmentation regions is units in rapidity so very large s is needed to make a clear �2-3 separation. Even at moderate s, though, one can study the ratio of the charge-difference plateau heights .

Because of the high energies available , and the possibility of controlling the momentum fractions x and ' systematic measurements of the hadron fragmentation region in a � the Drell-Yan process can lead to essential information on the complete hadron wavefunction in both the wee and valence quark domain. For example , the expected charge distribution in the valence region for is shown in Figs. and 7 . As in the K-p -t£-X 6 previous examples, we assume that the (uud) Fock space wavefunction is the dominant component in the proton at large ' and that the rapidities of the spectator u and d �j quarks are nearly equal to the initial rapi dity. If we now consider events with � small

147 or wee, diagrams Sa and Sb be-

come important (with a dominant

2 2 because Q = 4Q ). In models fragmentation u s region proton where the extra uii pair is crea- fragmentation 0 region ted from a neutral system, e. g. , 1�-logs-I from gluon decay as in Ref. 9, Fig. 7. The expected distribution of charge in then the wee qu ark charge is rapidity for the process of Fig. 6. compensated locally in rapidity,

K and one expects the qharge dis-

tribution shown in Fig . This 9. is also the prediction of the

"hole" fragmentation model of (a) (b) 10 4 Bjorken and Feynman. On Fig. 8. The Drell-Yan mechanism for - K p - in the "valence-sea" 1+1-x the other hand, the charge di s- region (xK 0. 2, 0. 2) . � �:S tribution of the proton fragments

may be more homogeneous,

since it is possible that the inter-

acting wee quarks are constitu-

ents of virtual charged and neu- - -1/3 770 tral mesons, as suggested in

Ref. 11. The same result is Fig. 9 . The expected distribution of charge in rapidity for the process of Fig . in predicted if we assume that the the "hole fragmentation" model. 6 bound quarks of the spectator

12 system exchange momentum and tend to equalize their velocities . The resulting charge distribution would thus resemble Fig. 10. Thus detailed measurements of the charge flow in the hadron fragmentation regions could well distinguish between these basic theoretical models. Comparison between the Drell-Yan process, production, and ordinary 1/J collisions should be illuminating .

- Other tests of charge and quantum number retention, especially in e+ e annihilation, are discussed in Ref. 3.

148 Fig. 10. An alternate possibility for the distribution of 1 / 3 + charge in rapidity for 71Q the process of Fig. 8 . 0 1. log (SXbj ) 3 III. A Model for Jet Fragmentation

One of the basic uncertainties in the quark model is the nature of the space-time evolution of the final state hadrons . constructing a model one must keep in mind that In

(aside from resonance decay) the emission of one hadron cannot cause nor directly influ-

ence the emission of other hadrons , since they are at a space-like separation. A simple

+ - model for e e -hadrons , which is a realization of Bjorken's inside-outside cascade

l O mechanism, is shown in Fig. 11. After the qq begin to separate , (virtual) gluons are

emitted with flat distribution in rapidity. One

assumes that each gluon lives, on the average , a

characteristic proper time T= 1/d, producing

quarks and antiquarks which recombine to form

color singlet mesons . The hadron production

2 2 2 then occurs near the hyperboloid, t -x = d , x=t=O which meets the quark world line at t - yd. The Fig . 11. Space-time evolution of the hadronic final state in e+e initial quark and antiquark are thus free for a annihilation. The initial qq pair is produced at x=t=O time t cc .fs which justifi es them as being treated and the hadrons are pro duced near the hyperboloid, as free particles in the calculation of the annihi 2 2 2 t -x =d . The transverse + - direction is not shown in lation cross section. In the e e center-of-mass the diagram . 13 frame, the fastest mesons are emitted last.

Although this model is grossly oversimplified, it is causal and covariant and has

many characteristics expected in gauge theories and jet production. Resonance and

baryon production can also be included. The model can serve as a simple testing ground

for the effects of quark mass and valence effects, etc. An application is the quantum

149 number retention rule, Eq. (2. 3) . Further, in this simple model the meson multiplicity is roughly equal to the gluon multiplicity and grows logarithmically. We discuss this feature further in the color model of Section VI.

IV. Leading Particle Behavior

In addition to its retained quantnm numbers an important discriminant of a jet is the x-1 behavior of its leading particles. The basic idea is as follows : consider a fast moving composite system A with a large momentum p //z . See Fig. 12a. The proba A bility that one constituent (or subset of constituents) a, in a virtual state , has nearly all the momentum of A must vanish rapidly as the remaining constituents , the n(iiA) spectators, are forced to low momentum . In fact, assuming an underlying scale-invariant 14 model one finds

2n(iiA) -I ----+ C(l-x) (4 .1) (x -1) where xis the light cone momentum fraction, x=(p�+p!)!(pt + p�) . This result for the 3 probability distribution leads to the predictions �G (x) � (1-x) , � (1-x) , vW2 p q/p vW 2rr G �(1-x) , G � (1-x) 7 , G �(1-x) 5 , etc. Comparisons with expe riment, correc- rr/q ij/p M/B tions and other modifications are discussed in Ref. 1.

An immediate application of the counting rule 1) is the prediction of the (4. c m. c.m. x =p . /p dependence of the forward inclusive cross section for the dissociation L z z max of a composite system. For example, one predicts for fast nuclear collisions (see Fig. 12b)

(4. 2) (a) and

3) (4.

Jl91A14 I bl since there are 3(A-A1) quark spectators forced Fig. 12. (a) Fragmentation of A into to low momentum . These and other predictions a subset of constituent a. The number of elementary have recently been shown in a beautiful analysis constituent spectators is n(iiA) . (b) Fragmentation 1 by Schmidt and Blankenbecler 5 to be in striking of A+B-a+X.

i 50 agreement with measurements of deuteron, alpha , and carbon dissociation at LBL. The 5 prediction G -(1-x) is also consistent with the Fermi motion observed in measure- p/d 2 1 ments of ed -epn for x=-q / p ·q-1. 6 2 d 17 One of the most intriguing applications of the counting rule (4. 1) is to forward in-

elusive hadron reactions pp - 7rX, etc. For x -1 this is normally considered the L 1-2a(t) domain of the triple Regge mechanism, where do/dx (A + B-C + X) - (l-x) . How 18 ever, in the range 0.2

At this point we shall distinguish three possible fragmentation models for high energy 19 hadron collisions : 17 (1) The incoming beam is dissociated by the Pomeron. Then as in Eq. (4 . 1) 2n(CA) -1 do/dx (A + B - C + X) -(1-x ) ; i.e. , the fragmenting jet is the excited hadron A. C (Although x is defined as the light-cone variable , x =x should be sufficiently C c R accurate .)

20 (2) Inelastic collisions begin with the exchange of a color gluon, as in the Low - 21 Nussinov model. See Fig. 15e1• The fragmenting jet is then an octet (A) with the 8 same quark structure as A. Again, one predicts the same distribution as in (1), C;iA. if (3) Inelastic collisions begin with the exchange (or annihilation) of a wee quark , as

in the Feynman wee parton model, as in Fig. 15e. The fragmenting jet then has one fewer 19 spectator compared to the gluon or dissociation mechanisms . We thus have

2n(CA)-3 do/dx (A +B-C+X) -(1-x ) (4. 4) C + + For example, for pp - 7r x, the 7r can be formed from the five-quark Jduuad> Fock-

state component of the proton. The Pomeron or gluon excitation models (1) and (2) then 5 give dcr/dx -(l-x) , corresponding to three spectators (dud) , whereas wee quark exchange 3 1 + / 1 (3) gives do/dx -(1-x) . The data 8 for pp - 7r X is consistent with (1-x · for !SR , c. m. 3 5 p <0. 5 GeV, 0.4

and VII.

i 51 Predictions for particle ratios are independent of which mechanism (1) -(3) is

assumed. For example, just by counting the extra quark spectators one predicts - + - (pp -K X)/(pp -K X) �(1-x ) and (pp -AX)/(pp -AX)� (1-x ) 8 which appears to be not K 4 L inconsistent with experiment�8 One also can predict ratios for different beams for mas-

sive lepton pair and production. These and other appli cations are discussed in Ref. 19 . ljJ the quark exchange or annihilation mechanism (3) is actually correct then there If can be a remarkable, long-range correlation set up in double-fragmentation reactions , For example in pp -7f7l) 7f 27 )x, with fast pions in the forward and backward direction, the requirement of quark exchange or qq annihilation forces an extra pair of spectators . One 3 then predicts da/dx (pp - 7f X) �(1-x ) (1-x ) 7 + (x -x ). This feature of the 1 ctx2 7f7l) 72) 1 2 1 2 model and further examples are discussed in detail in a recent paper by Gunion and my 19 self.

V. How to See a Gluon Jet

Thus far in this talk, the emphasis has been on quark and multiquark jets. It is

apparent that at some level QCD must imply the exi stence of gluon jets . Several essen-

tially scale-invariant processes have been suggested to find such systems. For example ,

the subprocess e e - -qqg (see Fig . 1 c') leads to a coplanar three-jet configuration, + 5 22 and the reaction y*q -gq leads to a double jet structure in the current fragmentation 23 region of deep inelastic electroproduction. However, the background from constituent + interchange model processes such as e e- -qqM and y*q -Mq is severe until very large pT and -ls ; this is discussed in detail by DeGrand, Ng, and Tye. 24 Gluon jets , of course, may also be predicted in high p reactions from gg -gg, Mq -gq, etc T . 25• 26 Recently Caswell, Horgan, and 126 have considered several processes which must have a lower limit for gluon jet production if ag'IO. For example, by comparing contri - butions the subprocesses qq -y* -µ µ for large p single muons and qq -yg for large + T pT (real or virtual) photons, we can derive a lower bound for scale-invariant hard photon production (see Fig. 13) , 26

1) (4. where e is the c. m. angle of the subprocess. This result should be applicable for - + - P GeV, where the qq -µ µ subprocess y ?, 4 dominates the single muon cross section.

Notice that all the uncertainties from the q (a) and q distributions cancel in this ratio. The

predicted cross section for a =O . and an s 25 estimated background from vector domi-

nance terms are shown in Fig. 14. When the

3191 Al5 ( b) subprocess qq -yg dominates , a gluon jet is + Fig. (a) Contribution of the qq- µ µ 13. predicted on the away side of the direct subprocess to µ+ production at large P · Contribution of y photon, although other subprocesses such as qq-yg to real(b) or virtual photon production at large P · y qg can also contribute here. -qy In general, any collision that produces

� direct (real or virtual) hard photons, e.g. , q\ PLAB 400 GeV/c \ + - -3 q - pp -y+X, e e -y + X, ep -ey+X, etc. , 2 \ Theory \ pp -yX (qq�ygl will also produce a gluon jet with a cross Q Experimental --0- - . 34 pp-rrXx 160 section from t h e sub st1tu. t· 10n a - 4 a . 26 A 3 s useful way to verify the hard photon-quark -36 + - current coupling in e e -qqy and eq -eyq

' ' is to measure the charge asymmetry in - 38 o_' + - ± ± ± e e -yh X, and e p-e yX as discussed in

Ref. -40 27 .

A gluon jet may be identified from the

2 3 4 5 6 7 8 9 global neutral character of its quantum num- P� (GeV/c ) 3 be rs, and relative to quarks , the suppression Fig. 14. Predicted contribution Edo/d p (pp -yX) at P = GeV /c, lab 400 of leading fragments. For example, one ec m = 90° from the scale invariant subprocess (qq -yg). expects G (x) /G (x) �a (l-x) log s/mq2 For reference , the Chicago 7T/q 7r/g s Princeton data of Cronin et al. where the nonscaling factor is due to the for pp -7TX is shown multiplied by a vector meson dominant -2 te2 falloff of the qqg- coupling . However, factor of 2 From Ref. 10- . 26. dkf/ ''T

153 the most important discriminant of a gluon jet may be its hadron multiplicity density dn/dy in the central rapidity region. We discuss this possibility in the next sections .

VI. The Dynamics of Color Separation

One of the central questions in the quark-gluon description of hadron dynamics is the question of what controls the magnitude and energy dependence of hadron production. In 12 a recent paper, Gunion and I considered the possibility that the multiplicity distribution at high energies depends in a quantitative way on the color separation initially set up in the collision. In quantum electrodynamics, soft photons arise via bremsstrahlung from initial or final charged lines , and the average multiplicity is computed from the sum over all charged-particle pairs, each contribution depending on the product of their charges and a function which increases with the relative rapidity separation of the pair. In the analogous case of quantum chromodynamics, charge is replaced by color, and the hadrons-which are color singlets-do not radiate . Radiation of colored gluons occurs only when two colored objects (e.g. , virtual quarks) are separated in rapidity. In addi tion there is a natural infrared cutoff determined by the size of the confinement region of color. We presume that the radiated color gluons eventually materialize as hadrons in such a way that thehadron multiplicity is a direct, monotonic function of the rising gluon multiplicity and hence only depends on the separating color currents. (A model where this relationship is linear is discussed in Section III.) Two processes with the same ini tial color-current configuration will thus produce the same multiplicity in the central rapidity region. (The principal effect of quark flavor will be to influence the quantum numbers of the leading hadrons .) The separation of color together with the eventual con finement of color thus leads naturally to a rising hadron multiplicity. + the canonical case, e e - -hadrons , the electromagnetic current produces at In time = 0 a quark-antiquark pair, and there is an initial separation of color 3 and Even 3. tually the systems are neutralized and produce hadron jets, as in th e model described in

Section III. It is evident from the structure of QCD that the gluon radiation depends on the magnitude of the color charge and the rapidity separation of the q and q systems , and that it is flavor-independent; i. e. , for the same rapidity separation the central hadron multiplicity is independent of which flavor quark pair is produced. particular , we In

154 + - expect a decrease of in e e -hadron at the charm (or other heavy quark) thres- had hold, but otherwise + =n (s) will have a smooth monotonic increase with e e- -had 33 log s.

This simple connection of color separation to hadron production implies that the

same function n (s) controls the central region multiplicity in every process which be 33 gins with 3-3 separation, e. g. , deep inelastic lepton scattering £p (£q -£q1) , and -f'X the Drell-Yan process A + B - fQX (qq See Fig. 15a, b, c. However, if a collision -ff) . involves the separation of other color charges, e.g. , color octet jets produced from initial gluon exchange or production of a gluon jet, then we predict a different, higher,

rapidity height. There is also the intriguing possibility of color interference when sev-

eral color systems are separated.

A crucial question in the above analysis is how to interpret the spectator system when a wee quark is removed from the hadron wavefunction. We shall assume that Fock

space quarks tend to have similar velocities and rapidities in a bound state , and thus at

the time of the interaction the spectator system can be regarded as a coherent 3 state 12 with the rapidity of the initial hadron. On the other hand, if one assumes that the wee

quarks are the children of gluons in lowest order perturbation theory,9 then the spectator

system would consist of a color octet at the rapidity of the hadron, and a 3 at the rapidity

of the struck quark . This argument seems tenuous , though, since the quark partons can

exchange gluons over the indefinite time before the interaction. The role of gluon

exchange in electroproduction is discussed further in the next section.

Given the simple 3 structure of the spectator wavefunction, we then are led to uni-

versal multiplicity plateaus in the current and hadron fragmentation regions of deep

inelastic scattering , and the prediction that the multiplicity in y*p is independent of - X 2 2 12 q at fixed s = (q+p) -even for real photons . These results are apparent agreement in 1 with experiment. 2 It is perhaps useful to briefly review the multiplicity calculations for QED. 8 The

analogous problem to the color situation is the calculation to all orders in of the num a + - + - ber of soft photons emitted by the outgoing muons in e e . The multiplicity is -µ µ given by a Poisson distribution where

155 expression

k max 1 l R = � - log .::..:..i:::.+ - 2 -elk (6 .1) 7T (3 1-(3 k 2 [ �jk . mm

4 2 1/2 Here (3= 1-m /(p · p _i is the relative velocity of the pair and y= � log is the [ + ] � relative rapidity. A rapidity plateau arises naturally from the k· p singularity of the

angular integration near the light cone , and the integration serves modify the dk/k to

height of the plateau. In QED, k . =O, and is logarithmically infinite. In QCD mm y where there is eventual confinement, the gluons of very long wavelength (k less than min 1 some hadronic size R- ) decouple since they only see an overall color singlet system and

the multiplicity can be finite . Since k < _fs/, y k s max = -2a log - - 1 log -- 2 k Y 1T ( m ) min

s � log log _- (6 . £1T 2....2 2) m k2 . mm where the hadronic k is the size of the color separation region. (In the model we mm-l. discussed in Section III this region grows in proportion to _fs.) 2 The result that the QED multiplicity (for k . O) behaves as log s at high energies mm f. is somewhat surprising and perhaps deserves further comment. We first emphasize that this contribution is not dominated by the photons produced at large k relative to the T 29 max charged lines. Even if we were to impose a cutoff at kT = k sine=k , the angular 2 2 2 integral still yields logarithmic form jde /(e + m /s) � log s. The actual transverse momentum distribution of the photons is interesting: the infinite sum in falls off as a a 30 Gaussian for moderate kf whereas at high kT , the scale-invariant dki/ki hard photon perturbation theory component takes over. Furthermore, the single photon distribution has a hint of the "seagull" effect. If we use light cone variables with x= (k k )/(p�+p;) , 0+ 3

1 56 y=log x C then·the leading + P ·P term in Eq. (6.1) gives + _

(6 . 3)

2 2 Thus grows with x m at small � · Notice thatin the central region (e . g. , x�o (m/ s)), the rapidity distribution is essentially flat, with dN/dy�a:/rr log s/k . . .f � mm '·--r Chromodynamics is of course much more subtle and complicated than electrodynamics , and all we shall do here is argue that QED at least provides a covariant and consistent model for multiparticle production which may represent a pattern for gauge theories.

We also note that the work of Ref. 31 suggests the possibility that the radiation of gluons in QCD may exponentiate into an effective Poisson form when gluons of the same order in the quark current are properly grouped together.

VII. A Two Component Color Model

An intriguing feature , evident from the perturbation theory structure of QCD, is that all reactions can be classified according to whether the initial interaction separates a 3

3 and of color or separates octets of color. This is illustrated in Fig. 15 for (a) electro

+ - production, (b) massive muon pair production, (c) e e annihilation, (d) high pT proc esses, and even (e) ordinary forward interactions. In each case only the initial interaction is shown; there is then subsequent gluon (and hadron) radiation which neutralizes the colored systems . 12 Using the color model discussed in Section VI, we expect all of the reactions on the left side of Fig. 15 to have the identical plateau height dn/dy for rapidities in the central region between the separated 3 and 3 systems. fact all of the jet parameters In , quantum number distributions , etc. should be indistinguishable in this region. In contrast to this we shall argue that whenever color octets are separated, the plateau height in the central region connecting their rapidities will be 2i times as high: -2 d /dy = 9/4 ctn /dy; the number 9/4 =2/( -n ) in color SU(n)] is derived from the 1138 33 [ l 12 lowest order perturbation graphs for gluon emission. Roughly speaking, an octet has a color charge equal to 3/2 that of the triplet.

We can also give an intuitive argument which shows why octet separation leads to a 32 rapidity height at least twice that of separating triplets. Consider the gluon exchange

157 3-3 SEPARAT ION OCTET SEPARAT ION

(a) (a') �' I"+ I" p p

p p

( b) (b')

H'3 �3 (c) ( c ' ) 3 p }3

M 3 3 p 3 ( d) (d ' )

p }3 8 q

3{ s{ e) ( e ' ) (

Fig. 15 . Contributions to electroproduction , massive pair production, e+e- annihilation, large hadron PT production, and forward hadronic collision, from both 3-3 and octet-octet separation in color . In (e), the qq can annihilate to a color singlet or a wee quark can be exchanged.

158 diagram in Fig. 15 d' for a large p reaction. It is clear that there must be two neutral T ization chains connecting the top and bottom 3 and 3•s, and the multiplicity will be double

that of electroproduction at the corresponding kinematics for 3-3 separation, as in 20 21 Fig. 15a. At low p , the same graph reduces to the Low -Nussinov model for the T Pomeron, Fig. 15e 1, and the two neutralization loops will interfere. The interference is not destructive since that would correspond to color singlet exchange. Thus the resulting + multiplicity plateau height must be at least double that of electroproduction or e e - -qq. + - + - The QED analogue of thi s result would be positronium +positronium -e e e e via photon

exchange. For a high p collision the soft photon radiation has the usual plateau height T along each outgoing lepton. At small p the interference of the radiation from different T charged lines causes the plateau height to vanish in the case of photon exchange, but

gives four times the height if the photon could transfer two units of charge . Notice that

in the case of the color, the interference effect vanishes if n - color 00 •

is possible that all hadron processes have both triplet- and octet-separation com- It ponents, but that the latter is suppressed, at least at low energies, because of the extra 12 associated multiplicity. Thus we speculate that quark exchange and annihilation gives

the dominant mechanism for massive pair production (Fig. 15b) (the Drell-Yan model)

low multiplicity large p reactions (Fig. 15d) (the constituent interchange model) , and by T 12 continuity to typical small p hadron reactions (wee quark exchange 4 ). (See also the T ' analysis of Section This ansatz , plus the color model, then can account for why the IV.) 2 multiplicity plateau is observed to be essentially universal, in all of these reactions.

On the other hand suppose we specifically consider events with high multiplicity,

e.g. , trigger only on events with at least double the usual hadron multiplicity. In this

case the color octet diagrams on the right side of Fig. 15 will be favored, exposing the 33 Berman, Bjorken, Kogut gluon exchange contribution for high p jets, and the Low - T Nussinov gluon exchange mechanism for low p hadron reactions. Furthermore, a new T + - essentially scale-invariant contribution from gluon exchange to pp -µ µ x shown in

Fig. 15b1 will be dominant. (Notice that, unlike the Drell-Yan mechanism, this contri-

bution gives the same production cross section for proton and antiproton beams.)

1 59 + - Similarly, the double multiplicity trigger in e e -hadrons will enhance the e+e- - qqg

contribution.

A two component color model for ordinary forward reactions automatically leads to

a correlation between left and right hemisphere multiplicities of the type observed at the

34 ISR : the gluon exchange component will be dominant for events with a large rapidity plateau height dn/dy are considered, thus giving a large multiplicity throughout the cen- tr al region.

Finally, we emphasize that by studying events with at least double the average multi plicity in pp collisions , one may be able to study qq -qq scattering as it gradually

evolves from the low p region (Fig . 15e') to high p scale-invariant jet production T T

(Fig. 15d'). This can provide a nearly bias-free way of determining the jet-jet cross

35 section. Aside from its dependence on the effective coupling constant a (p ) we s T emphasize that the gluon exchange term is scale-invariant, and thus unlike quark ex change , does not contain a strong p cutoff of the forward jets. T

Vill. Conclusions

this talk we have emphasized how the discrimination of various jet phenomena can In determine the basic quark gluon mechanisms which control hadron dynamics. In partic ular, we have shown how quark, gluon, and multiquark jets can be distinguished by their retained quantum numbers, the leading-particle x dependence, and the multiplicity plateau height. A summary of representative jet parameters is given in Table I. Mas

+ - sive pair production reactions A+B -JI. £ +X should be particularly interesting to study since the nature of the associated jets changes as one probes the wee and valence quark region. It is also interesting study these quark and multiquark jet systems in a to nuclear environment.

We have also emphasized the essential two-component nature of QCD and the rele vant role of quark- and gluon-exchange me chanisms. particular we have argued that In wee quark exchange is the dominant hadron interaction at present energies. The ansatz that gluon exchange and production contributions can be made dominant by using a double multiplicity trigger could be an important phenomenological tooL Its confirmation would estab lish the dynamical role of color separation in multiparticle production.

160 Table I

Jet Type Multiplicity Example Global Charge Leading Particle Typical Reactions

u 32 - 1J -/,(1- x) Q current induced, q n (s) large · 33 - { 1 � d -3-1JQ rr , (1-x) Drell- an

p, (1-x) qu"'k e�h""'e 4 { reactions , qq n (s) (uu) 3 + 1J 33 3 Q Drell-Yan, 3 /, (1-x) baryon spectators

2 M, (1-x) log s e e -qqg , pp -yX, gluon n (s) g 0 88 4 large pT , B, (1-x) log s r-current induced-

p, (1-x) gluon exchange in B n (s) (uud) 1 { hadronic reactions, 8 88 8 + 5 rr , (1-x) B + B-X

rr+, (1-x) gluon exchange in M n (s) cua) 1 { hadronic reactions, 8 88 8 5 p, (1-x) M + B-X

-l p, (1-x) diffractive B constant p 1 (1-x) A, { dissociation 5 rr+,(l-x)

-l rr+,(1-x) diffra cti ve M constant + 1 7f { dissociation K+ ,(1-x)

Acknowle�ments

Most of this talk is based on work done in collabo'.rations with R. Blankenbecler,

W. Caswell, J. Gunion, R. Horgan, and N. Weiss. I am also grateful for discussions with M. Benecke , J. Bjorken, T . DeGrand, F. Low, and H. Miettinen.

161 References

1. For a review and references, see S. J. Brodsky and J. F. Gunion, Proc. VIIInter

national Colloquium on Multiparticle Reactions, Tutzing (1976).

2 . The multiplicity data are summarized in E. Albini !:! �·, Nuovo Cimento 32A, 101

(1976) .

3. This section is based on work done in collaboration with N. Weiss. See S. J.

Brodsky and N. Weiss, Stanford Linear Accelerator Center preprint SLAC-PUB-

1926 (1977) .

4. R. P. Feynman, Photon-Hadron Interactions (W. A. Benjamin, New York, 1972);

Phys. Rev. Letters 23, 1415 (1969) .

5. G. Farrar and J. Rosner, Phys. Rev. D :!_, 2747 (1973) . See also R. Cahn and

E. Colglazier, Phys. Rev. D .l!.. 2G58 (1974) ; J. L. Newmeyer and D. Sivers ,

Phys . Rev. D .l!.. 2592 (1974); D. Novseller , Phys. Rev. D 12, 2172 (1975).

6. R. D. Field and R. P. Feynman, Caltech preprint CALT-68-565 (1977).

7. J. Vander Velde , presented at the International Winter Meeting on Fundamental IV Physics , Salando, Spain (1976), University of Michigan preprint.

8. S. D. Drell and T. -M. Yan, Phys . Rev . Letters 25, 319 (1970) .

9. V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, preprint

ITEP-112 (1976) .

10. J. D. Bjorken, Proc. Summer Institute on Particle Physics , Stanford Linear Accel

erator Center reports SLAC-167 and SLAC-PUB-1756 (1976). A. Casher, J. Kogut ,

and L. Susskind, Phys. Rev. Letters 31, 792 (1973) .

11. R. Blankenbecler, S. J. Brodsky, and J. F. Gunion, Phys. Rev. D 12 , 3469 (1975)

and references therein.

1 . S. J. Brodsky and J. F. Gunion, Phys . Rev. Letters 37, (1976) . wish to 2 402 I thank T. DeGrand for discussions on color interference phenomena.

13. Seealso F. Low, XII Rencontre de Moriond (1977).

14. R. Blankenbecler and S. Brodsky, Phys. Rev. D ..!.Q., 2973 (1974) . J. F. Gunion,

Phys . Rev. D 10, 242 (1974) . S. Brodsky and G. Farrar, Phys. Rev. Letters 31,

1153 (1973).

162 1 . R. Blankenbecler and Schmidt, Stanford Linear Accelerator Center preprint 5 I. SLAC-PUB...:1881 (1977). See also S, Brodsky and B. Chertok, Phys. Rev. Letters

37, 269 (1976) , and Phys. Rev. D 14, 3003 (1976).

16. R. Arnold et �· , Phys . Rev. Letters 35 , 776 (1975 ).

17. S. Brodsky and R. Blankenbecler, Ref. 14. See also H. Goldberg, Nucl. Phys .

B44 , 149 (1972).

18 . J. C. Sens �al. , quoted in W. Ochs, Nucl. Phys ., to be published. F. E. Taylor

�al. , Phys . Rev . D 14, 1217 (1976) . The p-.i\./p -A ratio on nuclei is consistent 8 or 9 with (1-x) . Overseth, private communication and W. Ochs, Rencontre 0 . XII de Moriond (19 77) .

19. S. Brodsky and J. Gunion, Ref. 1 (and to be published) . See also W. Ochs , XII

Rencontre de Moriond (1977).

20. F. Low, Ref. 13, and Phys. Rev. D 12, 163 (1975).

21. S. Nussinov , Phys . Rev. Letters 34, 1286 (1973) . See also J. F. Gunion and

D. E. Soper, UC-Davis preprint (1976).

K. 22. J. Ellis, M. Gaillard, and G. C. Ross, Nucl. Phys . Blll, 253 (1976) .

23 . E. G. Floratos, CERN preprint TH-2261 (1976).

24. T . De Grand, Y. J. Ng, and S. -H. H. Tye , to be publi shed.

25 . D. Sivers and R. Cutler, to be published.

26. S. J. Brodsky , W. Caswell, and R. Horgan, in preparation.

7. Brodsky, C. Carlson, and R. Suaya, Phys. Rev. 2 fl. D 14, 2264 (1976) .

S. Brodsky, J. F. Gunion, and R. Jaffe , Phys . Rev . D �. 2487 (1972).

28. D. R. Yennie, S. C. Frautshi, and H. Suura, Ann. Phys. 13, 379 (1961).

29. J. Dash, Marseille preprint (1977).

30. For a recent discussion, see G. Pancheri-Srivastava and Y. Srivastava, North-

eastern University preprint (1976) .

31. J. M. Cornwall and G. Tiktopoulos, UCLA preprints TEP 2 (1976) and TEP 10

(1976) . J. C. Taylor � al. , International Conference on High Energy Physics, XVIII Tblisi (1976) and J. Frenkel � al. , Oxford preprint (1976) .

163 32. Of course there is also the less interesting possibility that the result in QCD to all

orders in perturbation theory (unlike QED) resembles thermodynamic models

where the final state multiplicity has little to do with the initial process. wish to I thank Bjorken for conversations on this point. J. 33. S. Berman, Bjorken, and J. Kogut, Phys . Rev. D 3788 (1971). J. _!, J. Bjorken, Phys. Rev. D .§_, 3098 (1973).

34. See M. Benecke, Rencontre de Moriond (1977); and M. Benecke , Bialas , and XII A. S. Pokorski, Nucl. Phys. B lO, 488 (1976). wish to thank M. Benecke for discus l I sions on this point.

3 . For alternative method, see W. Ochs and L. Stodolsky, Rencontre de 5 an XII Moriond (1977).

164 GLUON SPL ITTING AND ITS IMPLICATION ON CONFINEMENT - A NON-PERTURBATIVE APPROACH*

York-Peng Yao Harrison Randall Laboratory of Phys ics University of Michigan Ann Arbor, Michigan USA 48109,

Abstract : We review briefly the perturbative treatment of the infrared problem and then argue that , for theories with self couplings, the situat ion in a non-perturbat ive approach is vastly different from that in QED. In particular, we explic�tly demon strate the impossibility to excite a hard gluon in because of successive splitting . In our view, this is equivalent(� ) 6 to hard gluon confinement .

Resume : Nous donnons un compte-sendu bref de la methode de perturbation du probleme infrarouge . Ensuite nous constatons que , dans les theories avec self-couplage, la situation des methodes non-perturbatifs est vastement differente que celle-la de QED. En particulier, nous montrons l'impossibilite de !'excitation d'un 3 gluon dur dans cause de "splitting " consecutif. A notre avis, celui-ci est(� ) 6equivalent, � au bornage des gluons durs .

*Work supported in part by the u.s.E.R.D.A.

165 I. INTRODUCTION

I would like to give a discussion of the infrared problem

relevant to the effort in understanding the structure of non ( l) ab elian gauge theory, wh ich is being actively pursued with the

promis ing outlook that it has to do with fundamental hadronic

interaction .

According to the lore , the non-abelian gauge theory pos-

sesses certain peculiar features. For example , it is argued that

wh ile the fundamental fields introduced are quarks and g luons ,

they cannot become asymptotic . Var ious conj ectures have been made ( 2) _to give theoretical justification for this confinement postulate.

They all are based on some assumed infrared behavior of the theory .

I would like to give a summary of a detailed calculation of the . 3 3 . ( ) infrared behavior of � in six dime nsions done by Chang and myself

and then advoc ate a specific mechanism for gluon confinement .

To lead to a self contained exposition of this subject, I

shall first give a brie f description of the problem and what has

been done in the past year or so, which bas ically showed that (4) there is order by order cancellation of infrared singularities

as in quantum electrodynamics QED.

Then I shall spend most of the time to g ive you some of the

results obtained by Chang and my self to show that for theories

with self couplings, such as nonabelian gauge theory, there are

distinct differences from QED. In particular, I shall explicitly

show that the true infrared behavior cannot be obtained via naive

perturbative approach. Finally, I will discuss the problem of

gluon confinement .

166 II . ZERO MASS PROBLEM

Now, what is the infrared problem - a better word is probably the zero ma ss problem - and why is it important for us to deal with it? Let me first state the problem. Essentially , the zero mass problem is a study of degenerate systems . When we are given a system with ma ssless particles , then there are two situat ions which can give rise to degenerate states :

( 1) Infrared : if we have a partic le with mass m and energy

E, then a state with this same particle together with an indefinite number of low momentum mas sless particles is degenerate in energy with it .

(2) Cbllinear : groups of parallel moving massless particles

+ in the same direction of total momentum P are degenerate in energy

i.e. if

-;. P2 ••• ••• such that n + p i�l then they all have the same energy n

i=l for an arbitrary n.

The study of degenerate systems is not a novelty in quantum

mechanics . The most relevent works in relation to our study here

are done by Kinoshita, Lee, and Nauenberg in the form of a theore(5)m.

What it states, in the field theory language, is that if the ma ss

renormalization constant in a theory is not singular in the mas s-

less limit of certain mass parameters , the n there exist ensembles

[i] , [j] such that the transition rates

167 2 l \ (i \ S \ j ) \ i d i] j dj] are finite . What this means in QED is that if we imagine the photon to have a mass for a moment , we must make sure that the electron mass is finite when the photon mass is made to vanish . The emsembles here are the we ll known Block-Nordsieck states.( 6 ) Here comes the importance o f such study : As far as the theory is concerned , these emsembles are the only states of any phys ical interest to us . Therefore, it is ne cessary to construct them first to form the state space be fore we can extract out the relevant physics.

168 III. PERTURBAT IVE APPROACH

There have been several demonstrative examples of infrared cancell,ations in low orders . Let me proceed to describe an example wh ich is very similar to QED and thus should be familiar to most. of us . This is the quark quark scattering .

Specifically, I take a set of quarks interacting with a set of non-abe lian gauge fields governed by a Lagrangian

c i 8 ab c _1__ 2

The e's are real and totally antisymmetric . The symmetry here is the color symmetry . The dynamics will be called quantum chroma- dynamics QcD.

For quark quark scattering , the situat ion is quite like that in QED . Recall that there the second order correction to electron

. e ectron scattering cross section. is . fin . ite . ,(7 ) if. we don't just 1 consider the effects due to virtual correction

eto . but alsoH soft emission H H � etc .

In words, what it means is that since it is ve�y easy to emit soft

169 photons, the probability of emitting no photon is zero; whereas if we allow soft photons to be emitted then we obtain a finite prob-

ab ility.

Note that in QED, because the photons don't carry any charge,

the initial and the final states have definite charge .

we now turn to QCD. It turns out that aside from a few

technical details, we can perform the same calculation for quark

quark scattering . However, in order to have a finite cross section we need to make an observation, namely that the color static charge

is gauge dependent and in fact an infrared divergent quantity.

Besides, gluons carry colors as we ll. Thus, for instance if we

assume the color symmetry to be SU ( 2) , then the quark quark

scattering cross section has both I = and I = 1 contents O

We do not obtain finite CT or CT rt is only the statistical I=o I=l· sum which is finite .

We have here a curious situation that the combination + + CT(pp+pp ) + CT (pp+ppp0) + CT(pp+pnp ) + CT(pp+npp ) which has the channel quantum number I = I, 1 =1, is infinite , 2 but the sum + + CT(pp+pp ) + CT(pp+ppp0) + CT(pp+pnp ) + CT (pp+npp ) - - + CT (nn+nn ) + CT(nn-�nnp0) + CT(nn+npp ) + CT(nn+pnp ) + - +CT(np+np) + u(np+npp0) + CT(np+nnp ) + CT(np+ppp ) + - +u (pn+pn) + CT(pn+pnp0) + CT(pn+nnp ) + u(pn+ppp )

+ +CT (np+pnp 0) + CT(pn npp0) wh ich has mixed channel quantum numbers , is finite .

170 From such simple examples , we can establish, among other

things, some results :

Coherent states do not have definite color contents and ( 1) only color blind rates are finite .

(2) Coherent states for a 'quark ' consist of the quark and an

indefinite number of soft gluons . (Block-Nordsieck)

Coherent states for a 'hard gluon ' consist of an indefin ( 3 ) ite number of parrallel moving hard gluons and an indefinite

number of soft gluons .

Static quantities, except for quark mass, such as color (4) magnetic moment , color charge , etc . cannot be defined on shell.

In particular, renormalization subtractions have to be done

off shell.

This program of perturbative finiteness to all orders have been pushed forward much further.

171 NON PERTURBATIVE APPROACH IV. Now I would like to describe some of the work being carried out by Chang and myself( 3in) the past few months .

We have seen that one can construct states such that infrared and coll inear singular ities cancel out order by order in QCD. In other words, there does not seem to be anything improper to assert that mathemat ically it is consistent to have these states .

What about dynamically? Can we produce such states? In particular, can we prepare a hard gluon?

Let me describe the following picture . If we start out with one gluon of some energy E > 6E = energy resolution, since it can cascade, we may end up with many many gluons . If this number is extremely large , then the energy of each one of them may be small enough to be below the energy resolut ion. They all become undetectable. If in fact the soft gluons always chew up all the hard energy this way, the mechanism is tantamount to gluon confinement if 6E can be made as small as we please. In this view the solf gluons are acceptable since they are needed to construct the coherent states.

One further question we want to pursue is this. Lately, there has been quite a bit of activit y in doing leading 0n sum calcul 8 atio� Jo investigate the infrared behavior of QCD . In such an approach, one expands a phys ical quantity A in power series of the coupling constant g

n=O and for each A ' one keeps only the most divergent term (usually n it is a power of as the soft cutoff is removed. We want to 0n ) show that such an approach is misleading and does not reflect the

172 true infrared behavior of the theory .

I want to use the following example to persuade you that we

cannot use naive perturbation to ob tain a meaning ful answer .

Let us consider the case when a quark passes by an external

singlet field . Because of acceleration, gluons will be shed .

Let \ M0 1 2 be the probability of the quark emitting one gluon with energy >6E , going into a color blind counter with angular

resolution 60

We can also find out the inc lusive probability of detecting an

arbitrary number of gluons by this counter of energy resolution

6E and angular resolution 60 . The calculation was actually

done by Chang and Tybursk�9 �n the leading 071 sum method . They concluded that the best way to do it is to go to the light cone gauge, where the diagrams which give the dominant contributions in each order are

+ �- --�� r" where + etc .

In fact, to obtain the dominant term in the probability , we need only the diagonal terms when we square the amplitudes, which give

1 2 � 2 -----=-----= I Mo 1 7 I Mo 1 2 1 c g 0716E 2 (1) ( - 07160 ) p

173 c here depends on the Casimir operator , and P is the momentum

trans fer .

This result is rather unphys ical. As we let 6E and 6ro�o , the inclus ive probability actually blows up first, because of the

pole , and then dies away . Phys ically it makes no sense whatsoever,

since as is lowered , we should be able to detect gluons in a 6E

wider spectrum and as we decrease 60 , we wo uld be ab le to see the

additional gluons which move closely to each other. Mathematically, it is also humbug because it involves such formal sum

m- 1 1 m x m=l

� in which the limit x � is of interest to us .

This last example is certainly a strong enough motivation for

us to perform a non-perturbat ive calculation to ascertain the true

infrared behavior . Now, because of gauge invariance, it is quite

impossible for us to truncate the non-abelian theory and obtain a

closed set of equations which we can solve , yield ing results which

can be trusted . What we do instead is to look at a theory which

has the general kinematics I described before : namely, we want a

theory which is asymptotically free , which has cascading effects,

and which has more or less the same singular structure in the soft

3 ( lO) emission limit . Th is is the � theory in six dimensions . Perhaps we should add a few words . It is not that we be l ieve

3 is a candidate for fundamental hadronic interaction . The (� ) 6 3 to understand the mechanism philosophy here is like using (� ) 4 for generating Re gge behavior in the high energy limit . Thus , we

place at the juncture more emphasis on finding a viable mechanism

than on quantitative agreements. As for the six dimensional

aspect, we may not find it so objectionable if we are reminded

174 that in QCD there is a derivative for each trilinear vertex . The

effective phase space volum e is six dimens ional .

QcD There is one difference wh ich I must point out, though. 3 has both infrared and collinear singularities , whereas ) has ( cp 6 only col linear singularities . As we shall see, this does not change the difficulty described earlier if we do leading I 0n sum.

Now the problem we post for ourselves is : What is the inclu-

sive probability for a virtual gluon of time like momentum p to dissociate into an arbitrary number of real phys ical gluons ?

would in the following establish a language which is dis- I tinct from naive perturbation, because it is in the way that we

can free ourselves from the old way of doing things .

Let me consider only trees . Then, the gluon field satisfies

. the c assica f ie. ld equation ( 11) 1 . 1 2 2 2 = A ( 2) (-o + µ lx g + 2 x

or 2 Gs + � G x x (3 ) where

G 1 (4)

have introduced a mass for the gluon field as a cutoff I parameter. I could do it with dimensional regularization . How-

ever, this way is more suitable for later discussion.

Now , to calculate the production probability

+ we can regard as a field operator and replace x ( ( + -) or ) Gs -.. i:p o cp o

175 - ( ) ( ) ± ± 2 X (±) + l.2 (x l ( 5) -cp0 rp6 is a free creation or an annihilation operator . The inclusive

dissociation probability is

F ( x-y )

(6) F p 2 p 2 ( ) <0 Using Eq ( 3 ) we have

= ) F (x-y ) (cp�+ (x ) cp�-) ( y ) ) ) ( 7 ( ) T (-\ } (1) 2 J dx1dy1G (x-x1)G ( y-y ) (x + (x x y1 + 1 1 > ( ) ( - ) We are going to expand (X + 2 x 2 ) into a series of AGF

the first few terms of which are

( 8)

It is best to represent this kind of expans ion in graphs .

Let me denote

( ) (-) + x y = -0 0------

Then the integral equation is of the form

4 + � - +

176 +

+ 0 8 (;\ )

This set of graphs is the same as th at we would obtain if we write down the proper self energy and make maximal unitarity cuts, i.e. retain nothing but the trees after the cuts . The vertices ( 1 ) are to be the bare ones, but the propagators are fully dressed. 2

If we keep only the first term in Eq. (8) and subsitute into Eq. and then Eq. we ob tain an integral equation (7) (6), 1

( 9 )

which can be written in the invariant mass variables s -p 2

177 d d 2 2 2 r o1 o2 s ( s ) µ 2 ( s -µ ) + F '/f o (_g:_)2'Tf 0 0 J 1 2 (10 ) 3/2 s ) 1"A( , o1 , o2 ( ) ( ) e o F o1 o F o vs -/o s2 1 2 2 1 .../0'2l where 2 2 2 g = / 84Tf fc 3 (11 )

and are the squares of the invariant masses of the cascaded o1 o2 blobs .

Note that because all the propagators are all time like in the tree approximation and we always have an even number of them in each term when we calculate the inc lus ive dissociat ion probability, our result due to truncation is a lower bound to the true probability .

we are interested in the limit of µ?0, keeping s finite .

However , because s and µ are the only two scales in the tree approximation , the limit is the same as s?oo but µ finite . We should nevertheless keep the true limit in mind , which is essential when we deal with renormalization.

Let me define

(12 ) and

= pf (p ) sF( s )

Then Eq. (10 ) becomes

(13) \ 3/2 . ( e ( p-q -q ) ) l 2 2s

178 We don 't know of any way to solve this equation exactly . If we are to iterate this equation in the coupling constant g and retain only the leading term in en (p/µ ), we obtain a result similar to Eq. (l) in QCD l sF(s) = 2 2 2 (14) (1 - }ir en(s/µ ) )

(leading sum) 0n the validity of which is being investigated , I will return to this 3 point later. For all it is worth , this also shows that (� ) 6 has similar infrared structure as in this approach. QCD We want to make the following observation : f(p) for 2 0 p because it is a probability. Therefore , if we replace the 2 0 3/2 kernel (A/ s2 ) in Eq. (13), wh ich is a positive semi-definite quantity in the allowable kinematic region, by another pos itive semi-definite kernel K(p, q ,q ) and call the corresponding solution 1 2 (13) f' (p) then 2 3/2 f' (p) f (p) if K (p, q ,q )<> (A/s ) 0 ?: l 2 2 (15)

we can easily derive the inequalities q q l 2 2 1 (1 - - 2 2 2 p (16) ..L.s -p l u b Hence the function f(p) is bounded from above by f (p) which

(17)

We define the Laplace trans form as

- bu r"' -"'p ub f = dp e � f (p) (18 ) (13) µ and

1 79 '( 19}

Then, we have a differential equation after the trans form

ub 2 dz -x 1 u b 2 -- = - e + (Z ) dx g_rr x (20)

The large p behavior is controlled by the small x region and vice

versa. Because of this, the boundary condition for the function ub z is that 2 ub -x z (x) e (21) + g_rr which is a statement that we should recover the one particle pole

for small p. When xis small Eq (20 ) is approximated by

(22) which has the solution

__ __ ub _ l z (23) {i;nX - {i;nXO We can show that the position of the pole 2 for small g (24) 2 for large g

Then, using Eq. (23) we take the inverse Laplace trans form to

obtain for p/µ >> 1 2 (x p/µ ) ub rr X o (25) F (s) e � pµO 2g 2 (14) Note that is non analytic near to g = Also, in this X O. O 2 approach, we can recover the result of Eq. ( 14) provided (g /rr ) 2 (p/µ) 1 {l;n < and g /rr << 1 . Now , let me construct a lower bound . call the solution by

the replacement q q \ 3/2 l 2 3 + (1 - + ) (26) ;.7 \ 1 p

180 lb F Its Laplace trans form satisfies the equation

2 lb = x 6 2 _g__ e - + (27) Tr 4x

We can show the following :

1 4 x ;; 0: b 140 (l/(en (x/x JJ ) z = 0 (28) and 2 2 0n = -Tr/g X0 for small g

= 2 2 /Tr ) - 40n(en (g /v ) J X &n(l for large g O (29) (28)' From Eq. we ob tain for p/µ >> 1 2 x 4 (x p/µ) - lb Tr 0 70 2 o F (s ) -- (p/µ) e (30) = 2 2g 3 µ

By assuming (x p/µ) n 0 F ) , p/µ 1 (31) (s = A p e >> (13), and substituting this into Eq. we can determine A and n for x • a given 0 We have in fact 2 1 3 sv 5/2 = (x /µ) , n 1/2 (32) A = 2 o g 3/27r

Because of Eq. (24) and Eq. (29) , we know that

2 for small g

2 for large g

(33)

It is obvious that the true behavior (Eq. (31)) bears no

(14) , resemblance to Eq. a result obtained by leading en sum method .

iBi V. CONJECTURE ON CONFINEMENT

I want to make an ob servation and a conjecture based on ( 31 ) (3 3 ). results of Eq. and Eq. From F(s) , we can calculate

the average mu ltiplicity, i.e. the average number of gluons which

the original gluon dissociates into

= 2 .iL (n ) g F s) 2 ( ;,g (34)

- 2 _a__ g 2 x p ( o /µ) (Jg SiEce

- 2 d TT g 2 x 0 e-0/ --2 4 g <>l g

we have 2 - TT ·-�-J /2 /g µ2/ 2 (n) 2 e-rr g << l g (36) I 1 2 "'! 1is- : / 2 \µ2) g >>l

we can ask a crucial question : What is the energy carried ,_µ? away by all the soft gluons with energy The answer is that

it must be greater than

µ TT 2 Lim ( ) "'! -TT g 2 n e / fs g l µ 70 2 << (3 7) g 2 '= rs g >>l

In both cases , it is a finite fraction of the total energy. In

particular, we are interested in the large coupling limit , because we are dealing with an infrared unstable theory. In this case ,

all of the energy of a hard gluon dissipates into soft ones, wh ich in fact are not moving at all. We can also calculate the

182 multiplic ity distribution , and a Gauss ian with almost zero width

results . This implies that the probability of having soft gluons

to chew up the hard energy is unity . If we agree that the soft

gluons are necessary components in constructing coherent states,

then we have provided a mechanism for confinement .

What about the quarks? I have not dealt with them so far ,

but we may envision a mechanism for their confinement . In order to

knock a quark out , we must kick it with something, which means

acceleration . The quark will then radiate gluons , which are all

soft and at rest relative to the quark because of the previous

cascading processes. It is possible that the effective mass of

this quark and the gigantic cloud of soft gluons around it becomes

practically infinite . The quark just can't come out !

I should contrast this with the situation in QED. There, the 2 Brems strahlung mean multiplic ity is (n ) � e w(p/µ), where pis

the momentum transfer and µ is the infrared cutoff. Therefore , 2 the mean energy taken by the soft photons is µ(n) � µ e £Jri(p/µ) ,

which can be made as small as we please by decreas ing the cutoff

µ. This , of course , agrees with our notion that electrons can

have well defined energy.

In the case when each gluon can split into r components, that

the multiplicity is not logarithmic can be seen from the following . (15) simple argument . Let e be the fraction of the original mass

which resides with each of the daughters in the form of rest mass

after each decay. Then, often n steps , each daughter has a mass n � e p, where p is the original mass. The process ends when n 8 p = µ. The total number of particles in the final state is n ( -wr/we ) � r : (p/µ ) . What we have shown is that in the strong

- coupling limit , 8 l/r . All final particles are then at rest with respect to the parent.

183 VI. REMARKS AND CONCLUS ION

We have given an explicit calculation, which shows that the

leading sum method does not reflect the true infrared behavior 0n of 3 or QC D. From the nonperturbative results, we have also (� )6 unravelled a dynamical feature, namely the impossibility to excite

any gluons of energy an infrared cutoff which can be made > µ , as small as we please . In our view, this mechanism is tantamount

to hard gluon confinement . We also believe that soft gluons , while experimentally undetectable, are nevertheless necessary

ingredients to construct physical states.

To be sure, we need many more investigations to find out whether this is the mechanism Nature chooses for confinement . .

Let me point out two : (1) What about radiative effects? The

folklore in that one consequence is to change the coupling constant

into a running effective one . In the infrared limit , this becomes

large . We have to do an analysis to see how it works in the time-like region . Let me point out one aspect here . We know

from low order calculation the reason that the effective coupling constant is large for small momentum (actually it has only been

3 shown to be small for large space-like momentum) in ( � )6 is due to vertex correction . Thus , to account for radiative effects , we need to look into equations of higher non-linearity than the one we have used so far , which contains self energy cuts only.

(2) What about the proposition that only color sing lets can be exc ited? We don't know how this is to come about, because color symmetry is not a natural symmetry in We need to turn to

QCD for an answer.

In any event, the mechan ism we propose here for confinement is a phys ical one . we hope that further progress can be made in

184 the near future to ascertain its truth .

Acknowledgments:

It has been my privilege to collaborate with s.-J. Chang on this sub ect . I have relied heavily on the results obtained us j by in preparing this talk . Encouragement from G.L. Kane is apprec iated .

185 FOOTNOTES AND REFERENCES

1. C.N. Yang and R. Mills , Phys . Rev . 96 , 191 (1954) .

2. For example, S. Weinberg , Phys . Rev . Letts . ll:_, 494 (1973) ;

D. Gross and F. Wilzcek, Phys . Rev . D9 , 980 (1974) ;

H. Fritzsch, M. Gell-Mann , and H. Le utwyler, Phys . Letts .

47B, 365 (1973) ; K. G. Wilson, Phys . Rev . DlO, 2445 (1974) ;

c. Callan, R. Dashen, and D. Gross, Phys . Letts. 63B, 334

(1976) ; K.J. Kim, unpublished .

3. s.-J. Chang and Y. -P. Yao , Non-Perturbat ive Approach to

Infrared Behavior and a Mechanism for Confinement , submitted

to Phys . Rev .

4. Y.-P. Yao , Phys . Rev . Letts . 2§_, 653 (1976) ; T. Appelquist,

J. Carazzone , H. Kluberg-Stern , and M. Roth, ibid 36, 7 68

(1976) ; L. Tyburski, ibid lJ..., 319 (1976) ; E. Poggio and

H. Quinn, Phys . Rev . Dl4, 578 (1976) ; G. Sterman , unpublished ;

T. Kinoshita and A. Ukawa, unpublished ; P. Cvitanovic,

Phys . Rev . Letts . lJ__ , 1528 (1976) ; C.P. Korthals Altes and

E. de Ra fael , Phys . Letts . 62B, 320 (1976) ; A. Sugamoto ,

unpublished ; etc .

5. T. Kinoshita, J. Math. Phys . (N. Y. ) l' 650 (1962) ; T.D. Lee

and M. Nauenberg, Phys . Rev . 133 , Bl549 (1964) .

6. F. Block and A. Nordsieck, Phys . Rev . �, 54 (1937) .

7. J. Schwinger, Phys . Rev . ]2_, 898 (1949) .

8. e.g. , J.M. Cornwall and G. Tiktopoulos , Phys . Rev . Dl3 , 3370

(1976) ; J. Frenkel, R. Me uldermans , I. Mohammad , and J.C.

Taylor, unpublished .

9. s.-J. Chang and L. Tyburski , unpublished .

186 10. Mack , Lecture Notes in Physics Vol . 300 (1972) ; Woo G. ..!:.2' G. and A.J. Mac Farlane, Nuc . Phys . B77, 91 (1974) .

11. D.G. Boulware and L.S. Brown , Phys . Rev. 172 , 1628 (1968) ;

Y. Nambu, Phys . Letts . 26B, 626 (1968) .

12 . This kind of equations was also looked into by A.M. Polyakov,

Zh . Eksp. Tear . Fiz . 22_, 542 (1970) [Sov . Phys . - JETP �' 296 (1971) l. 13 . We are not invoking any deep mathematical theorem in stating

this ob servation . We can prove it by simple iteration . For

example , if 3µ >p�2µ , the solution to Eq. (13) is

2 "' pf(p) =.9:.__ c dq dq vµ 6 (q -µ ) Vµ 6(q -µ ) • 2 1 2 1 2 'Tf µ 2 2 2 ;\(p q , q ) 3/2 ' l 2 , . ( (p-q -q ) 2 l 2 s 8

What we have claimed is simply that

2 pf(p) dq dq trµ (q -µ ) vµ 6 (q -µ ) e (p-q q J s; r"' l 2 1 2 c 2 g_2 µ o 'Tf (see Eq. (16) ). Now , the right hand side is the solution to

Eq. (17) for this range of energy . It fo llows that ub f(p) s; f (p)

We can construct a proof along this line for any finite µ and p.

The main point here is that we are considering a series of

sequent ial decays . By increas ing or decreasing the phase space

for each decay, we certainly should expect to obtain higher or

lower rate , respectively.

14 . It is amusing to note the instanton-like dependence on the

coupling constant. We presumably have included some classical

solutions in our approach , in a way which is not completely under-

stood by me . I thank H.-S. Tsao for a discussion on this .

15 . Th is is also known to A.H. Mueller.

187 188 THE INFRARED PROBLEM IN COLOR DYNf,MI CS

G. Tikco9oulos N.R.C. 11 Demol

ABSTRACT

Calculations of infrared singularities in non-Abelian ( Yang-Mills) gauge theories and in particular• Quar1tun1 C�hromodynamics are reported in the leading log.::i yii thm appr·oximat ion . Poss ible appli cations to harl.ron physics are discussed as explanation ot the quark rules tor exclusive fixed-angle such 1) dfl countin cro�,;s sectior1s , ir1c1icat ion of an infrar:::g d mechanism for quark confinemen-t an and a picturt:L) ot mCJ tter at lcirge interq_uaf'k distances resulting from hadr'onic the as33) u1nption of very strong range torces between colored quaDta . lor1g

i89 THE INFRARED PROBLEM IN COLOR DYNAMICS

INTRODUCTION

In recent years gauge theories have dominated theoretical thinking in parti cle physics and with good reason : they provided us with unified, renormalizable

1 models of weak and electromagnetic interactions ; and they also furnished us with a model of strong interactions such as Quantum Chromodynamics (QCD) in which

2 interactions between quarks become vanishingly weak at short distances - an explanation for (approximate ) scal ing in deep inelastic experiments . Furthermore , it seems that gauge theories are the only field theories capable of performing

3 these two feats - an indication that perhaps this time we are on the right track .

The dynamics of non-Abelian (Yang-Mills ) gauge theories with unbroken symme try such as QCD is at present far from well understood . Only recent ly , for instance , have we come to re alize that the structure of the vacuum state in YM

4 theories is considerably more complex than people ever had reason to suspect on the basis of their experience with ordinary field theories . Another serious complication is that the problem of infrared singularities is much more difficult to handle in YM Theories than in Abelian gauge theories like QE D. In these lectures , will describe work on the following three areas in which the infrared behavior of QCD has been examined in order to understand the physics of hadronic matter :

(1) Quark-counting scaling laws for exclusive cross sections at

high energy and wide angle and form factors at large momentum

transfers.

(2) A signal for quark confinement from infrared singularities .

(3) Hadronic matter for large interquark distances.

5 Sections 1 and 2 are based on work done in collaboration with J.M.Cornwall at UCLA . The oart of section 3 dealing with gluon chains is based on ref . 6.

All calculations and arguments presented here were based on the Lagrangian density of QCD:

i90 i wh ich couples a set of vector gauge fields B (gluon fields ) to a multiplet of

j fermion fields q (quark fields) . The matrices d are the group generators in

i i quark-field space and F is the gauge-covariant curl of the gauge field B µv µ

Here c are the structure constaniS of the gauge group i.e. SU( 3) of color i jk j and the matrices d are normalized according to

The Lagrangian can accomodate any number of quark flavors n,d,s,c, ... each

of wh ich appears as a color triplet.

1. ASYMPTOTIC FREEDOM IN HIGH-ENERGY WIDE-ANGLE EXCLUSIVE SCATTERING

7 A few years ago Brodsky and Farrar and independently Matveev , Muradyan and

8 Tavkhelidze pointed out that the experiments of exclusive high-energy wide-angle

hadronic scattering are compactly summarized by scaling laws of the form

2-N s f(t/s), s 00, s/t fixed ( 1)  :\; +

where s and t denote the squared c.m. energy and momentum transfer respectively

and

N total number of quarks of the initial and final

hadrons according to the usual quark model

assignments (meson � qq, baryon � qqq)

Moreover , the (spin-average d) electromagnetic form factor of a hadron for large

momentum transfer t behaves like

l-n F( t) � t ( 2)

191 where n is the number of quarks in the hadron . Thus , for example,

dCJ -8 � s for np dt TIP s -10 for pp pp

-1 F t TI -2 F t p

If these remarkable "quark counting" rules are true asymptotic laws for

hadronic reactions and not just a fluke of the present data , they probably re-

fleet some fundamental feature of the interactions between quarks at short distan-

ces . Can asymptotic freedom be invoked here ? Brodsky and Farrar noted that 2-N/2 Eq . implies that the relevant amplitude behaves like s and this happens (1) to be the behavior of any connected tree with N quark legs , provided the basic

interaction between quarks is renormalizable with dimens ionless coupling constants .

It could be , for instance , of the Yukawa type qq¢ with ¢ some set of "scalar

gluons" or an Abe lian vector-gluon theory qy qB or , of course, it could be QCD. µ µ The connected tree graph of Fig . is an example of a contribution to meson-meson 1 -2 scattering which , at wide angle , behaves like s (dotted lines are gluons)

meson{ � : } meson q

Fig . 1

meson {: q } meson q

To ge t the meson-meson amplitude this 8-quark amplitude must be appropriately

convoluted by the Bethe-Salpeter wave functions for the mesons . But this will

-2 ic,t change the s behavior if· the wave functions are sufficiently regular at

short interquark distances. The same is true for all other cases : meson-baryon

192 scattering, baryon-baryon scattering and form factors .

The question now arises : why isn't this tree-like behavior modified by loop diagrams ? What is the behavior of loop diagrams The behavior of scattering amplitudes at high energy and wide angle in field theory has been discussed by a

9 number of authors . The upshot of these inve stigations may be briefly summarized as follows :

"In all renormalizable field theories involving no elementary vector fields

(this excludes gauge theories the asymptotic power of s associated with a !) connected tree graph for a given process is modified by higher-loop graphs simp ly by multiplication with some polynomial in £n where is a renormalization �/1.2 /I. 2 2 point mass which is taken to be much larger than all particle masses : s » A >>m ". i s I. shall call these powers of £n //1.2 ultraviolet logs because they come from regions in the domain of integration of Feynman graphs where all internal momenta are of order at least A; or , equivalently , all vertices are concentrated in a

- 1 region of space-time of diameter /1. . They happen to be the same logs one finds for a given graph at large Euclidean momenta( if taken proportional to Thus , /S) . 10 for this class of theories, the renormalization group may be used to discuss the s + behavior of the exclusive cross section . In general, the resulting power of s would be given by the anomalous dimensions of the quark fields at the relevant

UV- stable point . There is , however , no � priori reason to expect that these anomalous dimensions vanish so that we get asymptot ically the same power of s as that of the tree graphs . Unless , of course, our theory is asymptotically free in which case all running coupling constants g( A) go to zero as /I.+00 and the trees would dominate . But , remember, for a field theory to be asymptotically free it must contain non-Abelian gauge (vector) mesons 3 and therefore it does not belong to the above category (i.e. the class of theories with only UV logs).

And so , to continue our search for an explanation of the quark counting rules within the framework of field theory , we turn to the study of the s + 00 limit in theories with vector mesons and , in particular , non-Abelian gauge theories .

It turns out that, in vector theories, in calculating the s -+ 00 behavior' of

193 individual graphs, one obtains powers of £ns not only from the UV regions we en-

countered before but also from regions where certain virtual momenta of vector mesons are much smaller than A (or s). A new mass scale is thus introduced which can be taken to be of the order of the vector meson masses if the vector mesons are massive , or , as in QCD , it can measure the departure of the momenta of all external colored particles from mass-shell. Such a departure is necessary because of infrared divergences of the mass-shell amplitudes. So in QCD besides the UV logarithms of the type £n( s/A2 ), there appear also infrared-induced (IR) 11 logari thms of the type £n ( s /M2 ). We may say that these logs come from domains IR where not all vertices of the graph are within a small distance of order A-1 from each other - some vector mesons are allowed to propagate by a finite distance of

- 1 order M

Since we are interested in discussing hadron scattering we must look at

"cluster" amplitudes in QCD (just like the one in Fig . 1) each cluster of external quarks representing a hadron . More precisely , we consider the s + 00 limit of connected cluster amp litudes (see Fig. 2) by taking the external momenta p as i follows :

p p = O(M) if i,j in the same cluster i j

n s, n .. fixed otherwise ij l] i 0

s + 00

(It) Cnl

Fig. 2

Pe Pg (NJ P,o (NJ

194 In Fig. 2, p +p is the 4-momentum of one of the pions and p -p = q is 1 2 1 2 the relative quark momentum which will be integrated over after multiplication by the pion BS wave function ¢(q) say , and similarly for all hadron clusters .

In what follows we will just discuss the s � 00 of these cluster amplitudes with

the p 's at off-shell values . We do not worry at this point about the on-shell i

divergence - we presume it is taken care of by the subsequent q integration and

the hadron wave function ¢(q) .

From a study of the asymptotic behavior of cluster• amplitudes in perturba-

tion theory there emerges , first of all, an important fact: if all clusters are

12 colorsinglets the IR logs cancel comp letely between graphs of the same order

This is quite analogous to the absence of infrared singularities in elastic scat-

tering of electrically neutral atoms . And since our hadronic clusters are always

color-singlet systems of quarks, it seems at this point that we can forget about

IR logs al together and that we have at least "explained" the quark-counting rules.

There is, however, one more complication . Right after the first suggest ion

13 of the power laws Eq. (1) and (2) it was pointed out by Landshoff that the

contributions of certain disconnected cluster amplitudes apparently dominate

those of Eq. (1) and (2). The simplest case is the one depicted in Fig. 3 for

meson-meson scattering .

Fig. 3

This class of graphs (appropriately convoluted with hadron wave functions)

gives a "pinch" contribution to the hadronic amplitude which behaves like

( 3)

195 where T 1 and T2 are the (connected) quark-quark scattering amplitudes shown in Fig. 3. What happens may be simply described in space-time language : quarks from different hadrons travel long (of order distances and collide at two M- 1) widely separated space-time regions (of size with amplitudes T and T re- A-1) 1 2 spectively. Since T , T themselves represent wide-angle collisions, unless they 1 2 -3 2 vanish in the s _,. 00 limit we would be faced with an s / behavior for the hadro- -2 nic amp litude which would dominate over the s behavior required by Eq. (1). Si- milar multiple-independent-scattering graphs exist for meson-baryon and baryon- baryon amplitudes.

So now we must investigate quark-quark scattering inthe s _,. 00 limit (at wide angle). Since we are no longer dealing with color-singlet clusters but with colored quarks the logs no longer cancel. Obviously, it would be interesting IR to know if they add up to something simple. Unfortunately , so far what has been done is a calculation of the leading logs in perturbation theory up to sixth IR order in the coupling constant g, for various processes . This is far from being complete or satisfactory . Nevertheless , something strikingly simple is suggested by these calculations : the logs begin to exponentiate just as in the more IR 6 familiar case of QED. One may state the result [true at least to O(g l] for a general connected cluster amplitude T as follows :

2 g 2 2 T T exp - {-�( E c )in Cs/M ) } (4) tree TI2 32 V V (only leading logs kept) where T is the tree approximation to T and c is the eigenvalue of the qua- tree v dratic Casimir operator for the representation of color SU(3) to which the v-th cluster belongs . For example , for the quark-quark scattering amplitude �cv=4cF where cF is the Casimir eigenvalue for the 3 representation of SU(3). Note , incidentally, that for color-singlet clusters , cv=O so Eq . (4) is consistent with our previous finding that there are no logs whatsoever (leading or non-leading) IR the clusters are color-singlets . if all.

So, finally , if we accept Eq. (4) , we see that T1 and T2 in Eq. (3) vanish

196 faster than any power of s and the Landshoff graph contributions are suppressed.

We may then conclude that asymptotic freedom is indeed the explanation for the 4 quark-counting ruleJ but only because infrared effects suppress the probability of scattering configurations which could require wide space-time separation be tween the constituent quarks. This looks like an intriguing hint - perhaps a related infrared mechanism is basically responsible for the strange fact that the quark constituents of a hadron never seem to fly apart to reveal themselves as free isolated particles . I shall pursue this idea and its possible relevance to quark confinement next .

2. INFRARED SINGULARITIES IN QCD �A SIGNAL FOR CONFINEMENT

In gauge theories like QED and QCD in which the gauge symmetry is not spontaneously broken, matrix elements of physical operators between asymptotic states containing charged or colored quanta are infrared-divergent when calculated in perturbation theory beyond the tree approximation level. Also exclusive cross

sections for gauge meson emission diverge at low momenta . In this section I 15 shall briefly review the infrared problem in QED and then point out certain apparent similarities and differences bet11een it and QCD - insofar as infrared

singularities in QCD can be organized and understood at present.

Begin with a simple process in QED like the one shown in Fig. 4. A

(virtual) neutral particle (e.g. a hard photon) represented by the dotted line + - produces an e e pair. The graph represents the exchange of a photon between the

outgoing charged particles. The "blob" amp litude T containsall the short range 0 interactions.

Fig. k 4

197 The corre sponding Feynman integral (assume scalar electrons for simplicity )

(2p 1+k ) (2p2+k ) T ------x o { ( P1 +k)2 -m2} {( +k )-m2 } P1 diverges at k=O logarithmically . One may introduce an infrared cut-off mass by replacing the photon propagator by (k2 -µ2 ). Then

Ii � a H(P 1P2 )£n(µ/M) x T � 0 ( 5) µ->-0 where the mass M is related to the inverse range of interactions in T0 (and not necessarily to the electron mass m) .

It turns out that if we add up all possible virtual photon exchanges

(including the no photon exchange T ) the result of Eq. exponentiates : 0 (5)

n T - I {aH£n µ/M) } T = T exp aH£n( µ/M) } (6) n Z -1,-n. ( o 0 { where { sum of In '" all n-photon-exchange graphs (Fig. 5)

Fig . 5

From Eq. (6) we see that T vanishes as µ->- O. This is not suprising because it can be understood clasically : one cannot observe the charged particles separated from their long range electromagnetic field. In fact, the energy density E(wl of classical radiation in this case approaches a nonzero constant as the frequency w goes to zero , and so we expect the number of photons E(w)/w to become infinite .

Now, if we ask fo� the probability to observe the charged particles plus an indefinite number of photons we should get a non-vanishing answer . This expectation

198 is borne out by a calculation of soft photon emission cross sections .

The graphs for the emission of one photon of momentum kµ have a pole at kµ=O (see fig . 6) 2

+···· Fig. 6

This implies a logarithmic divergence in the cross-section a1 as µ+ 0. We have

Q d 3k . Q T ·a i [ T 1i aHin (7) "' I 1 2 f (2pk )Z 2 < )Jl k' � 0 / 2 +µ2 µ+0 where Q is some fixed cut-off momentum (which defines what we mean by "soft" photon). Similarly , the n-photon amplitude has an n-ple pole at vanishing photon momenta , and it turns out that the corresponding cross-section on is given by :

.Q n [ T[2 � (aHin µ .) (8) % n.' µ+O

Thus the inclusive cross section is infrared finite because :

= 'f a n=O n

(9)

In fact , a is finite order-by-order in perturbation theory since expression

(9) is perfectly expandable in powers of a. This is a special case of a more ge 16 neral theorem: Lee and Nauenberg and independently Kinoshita have shown that in theories with massless particles (or more generally in quantum systems with infi- nitely degenerate energy levels) inclusive cross-sections are free of infrared

199 singularities order-by-order in perturbation theory .

Now let us turn to How much of what we said about is also true QCD. QED

in The evidence so far is based on leading-logarithm calculations in QCD ?

perturbation theory and even those are incomplete and scanty . Nevertheless, let

us begin by a process like that of Fig . 4 where a time-like hard photon (dotted

line ) turns into a quark-antiquark pair and a gluon is exchanged . The integral

is essentially the same as in QED ; we just replace in Eq . by c g2/4n . a (5) F

In higher orders of g, however , we encounter a much larger variety of graphs in

If we decide to take into account only skeleton graphs (i.e. graphs with no QCD. vertex and propagator ins ertions ) then it can be shown that the leading logs again exponent iate as in Eq . (6). The vertex and propagator insertions will almost certainly modify this (see next section) but we may assume here that any

way T will still vanish as µ � 0.

Next , we look at the one-gluon production cross section OJ.

The new feature of is that the emitted gluon is not "neutral" . Thus , it can QCD exchange soft virtual gluons with all other colored quanta (e.g. the quarks) as

illustrated in Fig . 7

Fig . 7 q

It turns out that processes with external gluons "on the ma ss-shell" i.e. with gluon momenta q such that q =µ2 , are more singular than those with just i f

(massive ) external quarks compared at the same order' of perturbation theory . The leading logs are powers of g2 £n2µ for gluon amp litudes (as compared to powers of g2 £nµ for quarks-only amplitudes ). These "double logarithms" are int imately related to the ones we encountered in our discussion of high energy , wide angle

200 limits (Section Again, up to sixth order the double logs for a multigluon 1) . amplitude exponentiate as follows (compare with Eq. (4))

( 10 )

The external gluons are grouped into clusters of one or more gluons each cluster having invariant mass of order µ. In Eq. c is the Casimir eigen (10) v +(v) . value for the v-th cluster and q is its 3-momentum. Going back to 01 now we see that , in contrast to the QED case, the exchanges of soft virtual gluons between the on-shell gluon of momentum q (see Fig. 7) and the other colored particles in the final state provide an exponential damp ing factor for the amplitude so that instead of Eq . (7) we now have :

( 11)

The integral in Eq. is finite for µ=O so that (11)

c 1 < cxi

Similarly, the n-ple poles of n-gluon-emission amplitudes do not cause loga-

rithmic infinities in a /a since the exponential damping factor (from soft n 0 (10) gluon exchanges) effectively only allows the emission of gluons or gluon clusters

with 3-momenta of order µ. Thus a /a approaches a finite value e as µ+O. n 0 n Assuming that the sum over n does not introduce a new kind of divergence, it

seems that the inclusive cross section o=La vanishes as µ+O at the same rate n as a • We interpret this result as a signal that in QCD quarks and gluons are 0 never produced in asymptotic states, i.e. as a signal for confinement .

Note that the suppression of gluons with three momenta much larger than µ

occurs only after summation to all orders of perturbation theory for each a and n

201 therefore there is no conflict with the Kinoshita-Lee-Nauenberg (KLN ) theoreml6

which states that the infrare d singularities cancel in the inclusive cross section

order-by-order in perturbation theory . It is really a matter of when the µ+O

limit of a is taken : if it is taken before summa tion to all orders of g then

the KLN applies , all tnµ 's cancel and we get a certain finite answer up to each

order of perturbation theory ; but if µ is taken to zero after summation to all

orders of g then a van ishes because gluon emission is disallowed.

Note also that the µ=0 value of the integral in Eq . behaves as l/g for (11) small g indicating that it is not possible to recover the perturbation expansion

for a1/a after µ is set equal to zero . 0

3. LONG RANGE FORCES AND GLUON CHAINS

In our discussion of the infrared singularities of amplitud.:s involving no

external gluons we simplified our task drastically by admitting only skeleton

graphs . By doing this we missed an important difference between QED and QCD

referring to the character of the long range forces between quarks .

In QED the behavior of the electromagnetic force at long range is related

to the transverse part of the photon propagator , which , for small values of the

-2 momentum q , behaves like q to all orders of perturbation theory provided all µ charged particles are massive . [This is because the vacuum polarization tensor

is (i) gauge invariant and (ii) has no charged-particle-produc tion threshold at

2 q = o] . Furthermore the infrared singularities of charged-particle vertices and propagators cancel each other because of Ward-Takahashi identities . This is

illustrated by Fig. 8 where (b) , (c) , (d) , (e) and (f) are the propagator and

vertex corrections to the second- order graph (a) for the process virtual

+ - photon + e e . For the present discussion it is more appropriate to introduce

the infrared cut-off µ as the departure of the momenta p and p' from mass-shell and not as a photon mass . Then as p+O , (a) is proportional to atnµ and (b) is pr�port ional to a2 £nµ because the insertion of the particle loop in the photon propagator does not make it more singular . The leading singularity of both (c)

202 2 2 and (d) is a in µ but it cancels between them; similarly for (e) and (f) . Thus the sum of (b) , (c), (d), (e) and (f) does not modify the £nµ behavior of (a) it just amounts to an infrared-regular charge and wave function renormalization .

We can express this by saying that higher-order quantum corrections do not modify

Co 1 ( c)

(d ) ( f)

Fig . 8

2 the l/r long-range character of the electromagnetic force .

The situation in QCD is different : the 1/r2 force associated with bare gluon exchange is modified by higher order corrections . Consider the process

+ - virtual photon�quark-antiquark pair (which is the analogue of y � e e in QED) .

In addition to the graphs of Fig. 8 (where the wavy lines are now gluons) we have

203 the graphs of Fig . 9. Note that the vacuum polarization now cons ists not only

of the quark loop contribution of Fig. Sb but also of that of the gluon loop

Fig. 9a and ghost loop Fig . 9b, both of which have threshold singularities at <2 ---<2 --- Cal Cb J Fig . 9 �--<:1 ---<1 Cc ) C dJ

Although the infrared singular contributions of individual graphs depends on the gauge , the ir sum (at least to this order of g) does not and it can be fully accounted for as a logarithmic modification of the small q2 behavior of the

17 transverse part of the bare gluon propa?ator in the graph of Fig. Ba. In terms of the polarization tensor we have :

ab 11 (q) µv

2 for q <

At present no one knows what these logs add up to and most likely perturbation theory estimates are not the way to explore the small q2 behavior of the gluon propagator , since we know that g=O i s an infrared-unstable zero of the

204 renormalization-group function S(g) . But these logarithms signal what could be

-2 a dr astic departure from the q of the gluon propagator To see what may be Dµv · hapenning , consider two colored quanta (quarks or gluons) r and s located at points x and x respectively . From the gluon propagator D v we derive an instan- r s µ taneous potential according to

(r)a (s)a t t U( ( 12) u rs I�r -�s I)

(r)a where t (a= l,2, ... ,8) are the matrix group generators for particle r. From

-2 l the bare gluon propagator (q2 )%q we get the Coulombic U%- l�l- , but if the D00 -2 full propagator behaves like (q2 ) , for example, we get U'.i:l�I . In this case the interaction energy of a quark-antiquark system in a color singlet configuration would grow linearly with their separation making it impossible to isolate a quark .

6 Gluon Chains

Although we don 't know what the small momentum behavior of the gluon propagator (more precisely : the effective gluon propagator , see above ) is , it is interesting to speculate as to what would happen if indeed it led to a very strong confining force at long distances. So assume that as a color-singlet quark - antiquark system is pulled apart the energy required grows linearly or faster with distance . Then it can be shown that it is energetically favorable to expend some energy to create a chain of gluons g ,g , ...,g joining the quark (q) 1 2 n to the antiquark (q) along some smooth curve :

The color-spins of the gluons must be coupled to the quarks so that the whole system is a color singlet (that's obvious ) and so that the system {q,g ,g , ...,g } 1 2 k

205 be longs to the 3 representation of color SU( 3) for all k. It is straightforward

to verify that this coupling in color space results in an exponential screening

of the forces between q and q. More explicitly, the color part of the wave function of the chains is

'¥ color

where a ,i,j are the color indices for the r-th gluon , the quark and the antiquark r a respectively and d is the a-th generator in the spinor representation . Using

Eq . we calculate the expectation value of the interaction energy between the (12) r-th and s-th quantum in the chain

<'¥ lu 1'¥ > col. rs col . rs - <'¥ I'¥ > col . col .

- l r-s l The factor (-8 ) due to the group theory of color is the reason for 1 having chosen this particular coupling in color space . Thus if I= O(m- Ii;.r +l-i;. r ) then lr-sl = O(mli;. I) and we have exponential screening of the long range forces . r -i;.s In a first estimate of the potential energy of the system we may neglect all

but the nearest neighbor interactions, and check the stability of the chain by

estimating its energy roughly by the relation:

in terms of a length R representing the spacing between gluons . Then each gluon

wave packet can have a spread of at most R and consequently an "uncertainty prin -1 ciple" kinetic-energy of at least R which accounts for the first term in Eq .(13) . -l The expression g2 (R)R is the interaction energy between two nearest neighbors .

The assump tions made about the gluon propagator imply that g2 (R) is large and - I 1 negative for R>>m and we also know that g2 (R) is small and positive for R<

(asymptotic freedom) . Then a stable minimum is attained for some value of R of

order m [a t which g2 (R)%0 (ll] . The energy of the chain is therefore proportional

to its length . Actually, the gluon spacing may vary along the chain so that the

206 energy content will be proportional to the length locally (in a coarse-grained sense ) .

The chain is stable against splitting in two pieces q,g , . ,g } and { 1 ,g2 . . k { g . . ,q} since the pieces, forming a 3 and a color systems respectively , k +l' " 3 would be subject to a strong long range mutual attractive force like the one between q and q (if un screened by the formation of a chain of gluons between them) .

The chain may split in two when a new quark-antiquark pair is created some- where along the chain and two color-singlet pieces q,g , ... ,g } and { 1 k ,q q,g , . ,q} are formed These may be easily separated since the strong long { k +1 . . . range forces do not act between them . Clearly, the chain may be split into more than two .pieces with the creation of more qq pairs at different points . We have here the picture of a meson decaying into mesons . Meson processes in general may be naturally visualized as events in which two chains combine into one (by the , inverse process to the one described above ) which subsequently is split in two or more color-singlet pieces . This description of hadronic matter is not much different from certain models of hadrons as relativistic strings with quarks at their 18 ends . What is different here is the coarse-grainedness of the chain and the specific role of color in its dynamics and stability.

Note that baryons may be described in an analogous way for large distances

one point between the three constituent quarks: three gluon chains are tied at to form a Y with the quarks at the free ends . The color part of the wave function should be :

a a a a a a 1 r r+l r+s r+s+l r+s+t e ( d d ) (d . . d ) (d . d ) i'j'k' ... i'i . j ' j .. ' k'k where the three chains consist of r,s at t gluons respectively with color indices a , ,a 1 ... r+s+t ; i,j,k are the color indices of the quarks . Again , this is the unique coupling in color space such that if the Y is cut at any point the two pieces have total color spins 3 and respectively . 3 It is instructive to realize that were the fundamental fermions to belong

207 to the adjoint representation of color (or any representation contained in a di-

rect product of adjoint representations ), it would be possible to screen the long range forces between say a fermion and an antifermion by surrounding each of them

with a small number of gluons (i.e. a number independent of the distance between

q and q) . A gluon chain would then be far from energetically favore d. Such

screened (or "bleached" ) fermions would probably behave like leptons toward each

other or versus ordinary hadronic matter (i.e. color-triplet quarks held together by gluon chains) . The possibility that physical leptons may fit this description

should be kept in mind.

The whole concept of gluon chains may be criticized up to this point as

based on the imprecise notion of an instantaneous color force and certain ad hoc

assumptions about its strength at long range . However , it should be pointed out that the zluon chain emerges rather naturally whe n one considers states created out of the vacuum not by gauge-variant operators like q(x)q(y) but by their gauge-

invariant (path-dependent ) generalizations :

Y a a - q(x)exp{ig d (z)d }q(y) ( 14) f "\J Bµ x,c

In Eq. (14) the line integral runs along the space-like curce C from x to y and the d matrices are ordered along C :

. n (n) - I

z n dz µ { (n-l) n-1· · ·

( 15 )

Since O creates meson-like physical states , a matrix element like c

where J is some local color-singlet current (e.g. the electroma- c 0 gnetic current) should be non zero . To simplify, take x �y �z and consider 0 0 io

for - large i.e,>>m c 0 \; y \ - 1

208 The n-gluon amplitude calculated in perturbation theory will dis-

play powers of g2£n [�-y [ which are the configuration space counterparts of the

infrared logarithms we described in Section 2 . We may assume that they exponen- n- tiate and make vanish as x-y -roo for any fixed n 0 I_,__,_I .

x x

� Z 1

Zr. 1

-- - J - - J---- � Z s-1

Z s

'------Y y

(aJ (b)

Fig. 10

However, for the band of terms in the expansion (15) with n � m[�-y [ the

situation is different . At the tree ·approximation level, the planar graph of

Fig. lOa has the color factor

a a a n 2 1 (d . . . d d ) . . ( 16 ) Jl

Consider ne xt the one-loop graph of lOb obtained from that of Fig. lOa by

exchanging a virtual gluon between the r th and s-th gluon lines . - If [!r -!s [ is large the loop integral behaves like £n[! -! [, an infrared singular behavior . r s n However , in calcu�ating the contribution of this graph to its color

209 (n) - r-s factorU7)combined with that of cr in Eq. ( 15) yields (-8 ) l l just as in the calculation of before Thus , all one-loop infrared logarithms like rs . �n - is large This l1r 1s l are suppressed provided l r-s l is large whenever 11r-1s l . is achieved by a configuration in which the distances between successive points

in the sequence x,z1 ,z2 , .•.,z n , y are kept bounded (below a common fixed bound) Such configurations are strikingly similar to gluon chains - look

specifically, at the coupling of the color spins given by (16).

Tree graphs which can be obtained from the planar one by a small number

corrections effectively suppressed by group factors and thus survive as contribu-

tions to the amplitude : these graphs have color factors representing minor local

"disordering" in the coupling given by Eq. ( 16) which does not disturb either the

screening of the color forces or the stability of the gluon chain . All other

graphs are infrared-suppressed This is worth checking in higher orders of per-

turbation theory .

In conclusion, I should emphasize that most of the work I described has its roots in perturbation theory and in graph-by-graph asymptotic estimates, although

the essential element be':l lr.d most arguments is exponentiation of infrared logs

wh ich presumes a summation to all orders . Eventually, of course , we hope to be able to study the infrared problem in Yang-Mills theories directly without having

to get our cues from perturbation theory .

210 FOOTNOTES

1. C.N. Yang and R. Mills , Phys .Rev . �. 19 1 (1954); a standard reference work

is the review paper by E.S. Abers and B.W. Lee , Phys .Rep . 9C , 1 (1973) .

2. Asymptotic freedom for non-Abelian gauge theories was shown by H.D.Politzer,

Phys .Rev.Lett . 1346 (1973) ; D.J. Gross and F. Wilczek, ibid. 1343 lQ_, lQ_,

( 1973) ; also G. t t Hooft , in Proceeding;of the 1972 Marseille Conference

on Yang-Mills Fields (unpublished) . For a discussion of the advantages of

Quantum Chromodynamics as a model of strong interactions see H. Fritzsch ,

M. Gell-Mann , and H. Leutwyler , Phys .Lett . 47B, 365 (1973) ; D. Gross and

F. Wilczek , Phys .Rev. DB ; 36 33 (1973) ; S. Weinberg , Phys .Rev .Lett . �.

494 ( 1973) .

3. For a discussion of the uniqueness of Y-M theories with respect to renorma-

lizability see J.M. Cornwall, D.N. Levin and G. iktopoulos , Phys .Rev .Lett . T lQ_, 1268 (1973) ; � ' 572 (E) (1973) ; C.H. Llewellyn Smith , Phys .Lett . 46B,

233 (1973); J.M. Cornwall, D.N. Levin , and G. Tiktopoulos , Phys .Rev . DlO ,

1145 (1974). Uniqueness with respect to asymptotic freedom is discussed by

S. Coleman and D.J. Gross , Phys .Rev .Lett . �. 851 (1973) .

4. A.A. Belavin, A.M. Polyakov, A.S. Schwartz , Yu.S. Tyupkin, Phys .Lett . 59B,

85 (1975); G. t'Hooft , Phys .Rev .Lett . �. 8 (1976); R. Jackiw and C.Rebbi , tunne lling Phys .Rev .Lett . �. 172 (1976) . Instantons and/ away from the zero-potential a (B =O) vacuum are ignored throughout these lectures , although some authors µ have argued that they may be essential to the problem of quark confinement .

5. J.M. Cornwall and G. Tiktopoulos , Phys .Rev .Lett . �. 338 (1975 ); Phys .Rev .

Dl3, 3370 (1976).

6. G. Tiktopoulos , Phys .Lett . 66B, 271 (1977).

7. S.J. Brodsky and G.R. Farrar , Phys .Rev .Lett . �. 1153 (1973) ; Phys .Rev. Dll,

1309 (1975) . For a recent review see D. Sivers , S.J. Brodsky and R. Blanken-

becker , Phys .Rep . 23C , 1 (1976).

8. V. Matveev, R. Muradyan and A. Tavkhelidze , Lett . Nuovo Cimento 719 (1973) . '}_,

211 9. See G. Tiktopoulos , Phys .Rev . Dll , 2252 (1975) for a rigorous discussion to

all orders of perturbation theory for a ¢4 interaction . The following

authors have independently arrived at the same conclusions : M. Creutz and

L.-L. Wang, Phys .Rev . l.!:_, 3749 (1975 ); C.G. Callan and D.J. Gross , ibid. , l.!:_, 2905 (1975). The large momentum transfer limit of form factors in non

vector field theories has been discussed by G.C. Marques , Phys .Rev . D9 , 386

(1974); S.-S . Shei, ibid. l.!:_, 169 (1975) ; J.M. Borenstein , ibid. l:.!:_, 2900

( 1975 ) .

10 . M. Gell-Mann and F.E . Low , Phys .Rev. �. 1300 (1954); C.G. Callan , Jr .,

Phys .Rev . D2 , 1541 (1970); K. Symanzik , Commun . Math .Phys. �' 227 (1970) .

11. The powers of in (s/M2 ) are called infrared-induced because they connect the

on-mass-shell (M=O ) divergence of the Feynman integral with its large s

behavior .

12 . The detailed proof will be included in a forthcoming publication . In re f. 5, only the cancellation of the leading logs had been checked .

13. P.V. Landshoff, Phys .Rev . DlO , 1024 (1974); see also J.C. Polkinghorne ,

Phys .Lett . 49B , 277( 19 74 )fcr the suggestion that the Landshoff-graph contribu

tion might be suppressed by infrared effects in the context of an Abelian

gluon mode 1.

14 . Strictly speaking, since the running coupling constant approaches zero, even

the tree-graph contribution should eventually become negligible at high

energ¥ compared to the completely disconnected graphs, i.e. those without

any large-momentum- transfer gluons whatsoever . This is the "constituent

interchange model" of R. Blankenbecker , S .J. Brodsky and J.f. Gunion , Phys .

Lett . 39B , 649 (1972); Phys .Rev . DB , 187 (1973) .

15 . The physics of the Abelian infrared problem (QED) was first correctly describ

ed by F. Block and A. Nordsieck , Phys .Rev. �' 54 (1937). A comprehensive

treatment of infrared logarithms is given by D.R. Yennie, S.C. Frantschi ,

and H. Suura , Ann .Phys . (N.Y.) -1:2_, 379 (1961) , and G. Grammar and D.R.Yennie , Phys .Rev . ..!!.§_, 4332 (1973) . For a highly readable account which also includes

212 infrared gravitons see S. Weinberg, Phys .Rev . 140 , B5 16 (1965 ).

16 . T. Kinoshita , J.Math .Phys . 2• 650 (1962); T.D. Lee and M. Nauenberg , Phys . Rev. �. B1549 (1964). For a discussion of the validity of this theorem

in Y- M theories see E. Poggio and H. Quinn , Phys .Rev. Dl4 , 578 (1976).

17 . J.M. Cornwall and G. Tiktop�ulos , "Infrared Behavior of non-Abelian Gauge

Theories II" , UCLA preprint ( 1976). Bars and A.J. Hanson , Phys .Rev . D13 (1976 ); W.A. Bardeen , Bars , A.J. 18 . I. I. Hanson and R.D. Peccei, Phys .Rev. D13 (1976).

213 214 UNIVERSALITY OF QUl\RKS l\ND LEPTONS

H. FRITZSCH

CERN -- Geneva

l\BSTRACT

T.1e unification of QCD and flavour dynamics of quarks and leptons within unified theory based on a a simple gauge group is discussed . Two possibilities ("or thogonal11 unif ication, 11 ex ceptional." unif ication) are especially studied .

21 5 1. INTRODUCTION

Now it is almost ten years since the first results of the electroproduction experiments at SLAG became available for theoreticians to think about . Those ex perimental results, and the results of the subsequent neutrino production experi ments, have made it clear that quarks do not only play a fundamental role in hadron spectroscopy but that they also behave as nearly point-like objects inside hadrons, just like leptons ins ide atoms . Thus quarks and leptons are very much al ike , and one might speculate if in some sense quarks and leptons are different editions of the same basic entity . In this talk I shall discuss some of the ideas theoreticians have discussed in order to realize such a universality of leptons and quarks .

Let me first describe the present picture of particle physics which has emerged during the last few years . We have reasons to believe that all interactions ob served thus far belong to the following three clases of gauge interactions :

I Gravity (gauge group : Poincare group) colour] II QCD [gauge group : SU(J) III QFD (quantum flavourdynamics: weak, electromagnetic and possibly other interactions , gauge group yet unknown) .

In particular we are dealing with three types of unbroken gauge interactions (gravity, QCD, and electromagnetism) . On the other hand , QFD is described by a spontaneously broken gauge theory . The smallest possible gauge group of QFD is SU(2) U(l) , and the minimal set-up of elementary fermions is described by the x following scheme :

u (1.1) g d (:� g where the 1nd1ces r, g, b refer to the quark colour (red , green , and blue) . The eight basic fermions denoted above are those fermions which constitute the ob served stable particles in the wor ld . The scheme (1. 1) is not self-consistent , due to the existence of the Cabibbo angle. The latter provides a bridge to the next layer of heavier basic fermions, and the minimal set-up of elementary fermions consistent with present observations is given by the familiar four-quark/four-lepton scheme

u li g b d' d') v g b (:� d' d cos 8 + s sin t� c e c (1.2) s' � c g s ' s'Cb) c�v g b (��

216 Until recently it looked as if this scheme describes all basic fermions in the world. However, as we heard, for example, in last week's session on lepton physics, the existence of another heavy charged lepton with mass just below 2 GeV seems rather certain; thus one should be prepared to expand the system of elemen tary fermions even further .

2. QUESTIONS

What are the puzzling questions theorists come across by looking at the emerg ing picture of basic interactions? One question which came up first, with the discovery of the proton, is the question of charge quantization. Within the con ventional SU (2) U(l) framework the experimental fact that the proton and positron x charge are equal , or the charges of the u and d quarks are 2h and -1h respectively, is a complete mystery.

Ano ther puzzling question is the question which constraint dictates the colour colour and flavour structure of the elementary fermions. Why SU(3) , and not Z colour colour SU ( ) , or SU (4) ?

In the conventional picture (see above) the gauge groups of the three basic interactions commute with each other , certainly not a very appealing situation. Can we envisage a situation where those three groups emerge as subgroups of a large simple group?

Another important question is: What kind of physics dictates the mass pattern of leptons , quarks , and bosons? In the conventional gauge theory framework all masses are generated by spontaneous symmetry breaking . However , in all schemes discussed thus far by theorists the number of free parameters is comparable to the numb er of mass parameters; no mass relations exist. The various masses have to be adjusted ; clearly a highly unsatisfactory situation. Who can do better?

3. ATTEMPTS TO QUANTIZE ELECTRICITY

3.1 quantization within QFD

The most attractive way to obtain charge quantization would be to realize it within QFD , for example within the system of the observed weak and electromagnetic interactions . Unfortunately the pattern of those interactions as observed in nature does not allow us to realize such a possibility: The weak and elec tromagnetic in

teractions observed seem to follow an SU(2) x U(l) pattern. Can we embed the group

SU(2) x U(l) in a larger simple group , in which case the charges would be quantized? The simplest way to do it is to use the group SU (3) . However , in the conventional

picture of QFD based on the scheme (1.2) the fermions do transform under SU (2) x U(l) as SU (2)-doublets or SU(2)-singlets in a way which does not allow an extension to

217 SU (3) . An expansion of the fermion system is necessary. In Ref . 1 the minimal , expansion of the fermion system was discussed . What is needed are two new quark flavours of charge -1h (b,h) , two new charged leptons (E- ,�) , and four new neutral weak leptons. The scheme is vector-like , the SU(2) representations are:

(3.1) (b" d') (h" ) L R SR' ' ' -L '

(b11 = b cos 811 + h sin 811, h11 = -1,81 1: right-handed analog of the Cabibbo angle) (L : left-handed , R: right-handed)

N' N" oL (.'. ,�\ (:;1 oR e \ µ JR E JR

(N� =N cos a + N sin a, N� a: leptonic weak mixing angle, N ,N : massive e µ = .L; e µ neutral leptons , m m mK; N: = N cos + N sin Ni N� = N cos + Ne • Nµ <: 3 fl' 0 S', = -1; 3 S" +No sin 6 1 triplet-singlet weak .mixing angles , N ,N : neutral leptons , S";: ,S"; 3 0 masses unconstrained .)

The fermion scheme (3.1) can be arranged in an SU(3) scheme as follows : • • N'µ µ Vµ µ

I \ I \ I \ I \ I \ I \ N ' I N" \ II \ J ' I •3 ' E-( ) M• E-( ) M+ • • \ I \ I ' N' I \ N" I ' 0 I \ 0 I \ I \ I ' I ' I e- (3.2) e- Ve N'e L R

' d' u s c b" u h" c \ I I I \ I \ \ I \ I \ \ I \ I I \ I \ I \ I I \ ' I \ I \ I \ I \ ' I \ I \ I v v v' v b" h" d' s ' L R

218 The gauge group of the observed weak and electromagnetic interactions is re garded as the subgroup SU(2) x U(l) of SU (3); the fermions transform under colour flavour SU (3) x SU(3) as follows :

quarks (3 .3) leptons

Within the SU(3) scheme the charges are quantized: The integral charges of the leptons and the fractional charges of the quarks are obtained as a consequence of the octet structure of leptons and the triplet structure of the quarks. The 2 3 SU (2) x U (l) mixing angle 8w (unrenormalized) is 60° (sin 8 = £), i.e. the neutral current is a pure axial vector. This is excluded by experiment ; thus sizeable re 2 normalization effects must exist such as to give sin 8w $ 0 .6 which is required by experiment (for a detailed discussion of the phenomenological constraints see Ref. 2) . The masses of the SU(3)flavour octet bo?ons, except the ones belonging to the SU(2) x U(l) subgroup of the conventional weak and electromagnetic inter actions, must be very heavy implying that the interactions caused by them are negli gible for present day phenomenology. 3.2 Leptons as the fourth colour Another way to obtain charge quantization is to enlarge QCD by including leptons as "the fourth colour", following an idea of Pati and Salam3). [Note, however , that those authors base themselves on unconfined colour [sU(3)colour is a broken gauge group] , and the quarks are assumed to have integral charges. I shall follow the route described in Ref . 4.] Of course, the leptons are not really a fourth colour , in which case they would be confined like the quarks. What is meant is that SU(3)colour is an unbroken subgroup of SU (4)c , the latter being a broken gauge group . The fermion scheme can be described as follows :

u u g b ""' d' ; d' o g b J :� (3 .4) c�.. "V"" su4 [and an analogous set-up for the (c-s' ; system] . Vµ-µ-) c 1 R Suppose we start out from the gauge group SU(4) x SU(2) x SU (2) , where SU (2)L(R) are groups under which the quark and lepton flavours (left-handed, right-handed respectively) transform as doublets. (Since no right-handed charged weak currents are observed , the boson WR+ must aquire relatively high masses) . The electric charge can be written as

219 2/3 2/3 2/3 I 0 1 I 1 1 1 L+R ( = + T + y -g ( 3 � 1/3 -1/3 -1/3 '! -1) i (1 -1 -1 i'J-1 1 (3.5)

where Y is the so-called lepton-quark hypercharge [it is the 15-generator of SU (4).] . The interesting feature of Eq. (3.5) is that the electric charge is the sum of two generators of non-Abelian groups; thus the charges are quant ized . Note, however, that there is still one flaw in Eq . (3 .5) . Since we are deal ing with the semi 1 R simple group SU (4) x SU(2) x SU(2) , which allows two independent coupl ing con stants , if we impose 11parity invariance" of the Lagrangian, there is in general an arbitrary parameter in the expression of the electric charge . In general we have

L+R T + a . Y (3 .6) 3

e e the ratio Q (u-quark) /Q (e-) is no t ( 2h)/-l , but [ (3 a)/(3 3a)]/-l , and the neu + + 1 R trino charge is not zero , but (1 - a) . Thus within the SU (4) SU (2) SU (2) 1h x x scheme the charges come out correctly only if we impose in add ition Q(Ve) = 0 (a 1) . c colour The fifteen gauge bosons belonging to SU (4) transform under SU (3) as ( 5) = + + + l; gluons , (�) : leptoquark bosons , Y. ! (§) (2) (2) (§) : (�) , (�) : The present phenomenological constraints on the leptoquark boson ma sses are not ) very strong (� 100 GeV) . Leptoquark exchange can lead to proton decay 3 , in which case they have to be slightly more massive as ind icated above . However, proton c decay is not a necessary consequence of the SU(4) scheme . There exist ways to break the gauge symmetry such that baryon number is exactly conserved (see Ref . 4 and Ref . 5) .

4. UNIFICATION OF QCD AND QFD

The ultimate goal of physics would be to unify all interactions by some per haps relatively large simple gauge group . Today not even an ansatz in this direc tion exists; all attempts made in this direction have failed due to the enormous conceptual difficulties to incorporate gravity. Thus the best we can do at the moment is to leave out the gravitational interaction, reserving to include it at some later time , and try to construct theories which unite QCD and QFD within a simple group . Many models have already been constructed . There is no point to review all of them here . What I will do instead is to describe two canonical ways to achieve a unification, both based on the two different schemes discussed in the previous section.

220 Let me first mention one feature of unified theories wh ich is of phenomeno logical importance and does not depend on the specific group . If we choose a sub

group SU (2) x U(l) of a simple group and declare it to be the gauge group of the

observed weak and electromagnetic interactions , the SU (2) x U(l) mixing angle is 4) fixed to

2 sin 8 w (4 .1)

where the summation is to be carried out over all left-handed and right-handed fermion fields (including colour) . For example, in the scheme (1.1) and (1.2) one

obtains sin2 8 = The resul t (4.1) describes the SU (2) U(l) mixing angle w 3� . x in the energy region where the unification becomes relevant. Thus it cannot be applied blindly for phenomeno logical purposes , since relatively big renormaliza tion effects can occur . There exist theories where the latter are really substan- . . 4 5) tia 1 , wh'l 1 e 1n. ot h er t h eor1es . essent1a. 11 y no renorma l'1zat 1on. 1s present • •

4.1 Orthogonal unification L R c x Let us return to the SU (4) SU (2) x SU (2) scheme discussed in Section 3. How can we unify this direct product of three groups such that the correct fermion pattern results?

Let me use the identification SU (4) S0(6); SU(2) x SU(2) 80(4). Thus we � � 2 4 ) realize one way of embedding SU(4) x SU (2) in a simple group

( 4.2) x x x c: so10

Do we obtain the desired fermion content? Yes , if we choose the complex (16) re- c -L R presentation of SO (lO) , which decomposes under the subgroup SU (4) x SU (2) x SU (2) as follows

(16) (4, 2, 1) 1, 2) + (4, (4. 3)

These are one quark-lepton doublet and the corresponding antiparticles. The fermion scheme (1.2) can be described by two (16) representations of SO (lO)

221 ' I 'V c u u IU u c c c ,ve 1Jµ I I I : d' d' l e s' S (: . ' d' d' s' s'

Note that right-handed neutrinos (left-handed antineutrinos) have to be added . The breaking of the SO(lO) gauge synnnetry can proceed in two different ways, given by the following diagram

SO(lO)

SU (4)c x SU(2)1 x SU(2)R SU (S)

SU(3)c x SU(2) x U(l).

The subgroup SU(S) results, if we go along the chain on the right-handed side; the (16) decomposes under SU(S) into: (!6) = C!O) + (�) + (!) . In this way we arrive at the SU(S) theory of Ref . 7 as a subtheory of the SO(lO) scheme . 4.2 Exceptional unification c flavour What kind of unification can one discuss for the SU (3) x SU (3) scheme discussed in Section 3? We must look for gauge groups which provide "natural" embeddings of SU (3) groups . Such groups are the exceptional groups , and we shall study below various lepton-quark schemes based on the exceptional groups (G2 ,F4,E6 ,E7 ,E9) . The pioneering work in this direction has been done by s the Yale group ) . For details on the group theory aspects we refer to the mathe 9 matical literature ) . The group G2 This group being of rank 2 is, of course, too small in order to be useful as a gauge group for a unified theory of the strong , electromagnetic, and weak interactions. Nevertheless a gauge theory based on G2 is an interesting example

222 since one can display here in a simplified manner some of the problems one is dealing with in case of the other exceptional groups. The group G2 has as its maximal subgroup of maximal rank the group SU(3) . Let us identify this group with the colour group SU(3)c . We suppose that the gauge c group G2 is broken such that SU(3) is left as an unbroken subgroup

sue3 c The basic representation 7 transforms under SU(3) as = 3 + 3 i.e. it con- 7 + 1, sists of a quark triplet q, antiquark triplet q and a lepton. (Majorana lepton.).

The adjoint representation decomposes as = + 3 + 3. Besides the colour genera 14 8 tors (coupled to the gluons) one has six generators which have both the quantum numbers of a diquark (qq) and a leptoquark (qi) , i.e. a diquark can annihilate into an antiquark and a lepton via the diagram displayed below:

b �>---<__ :_ _

This feature (which is a general feature of the exceptional groups) implies not only that there exists no baryon number generator in the G2 scheme , but that a three quark colour singlet system like the proton can decay in second order of the gauge coupling . The only way to suppress this decay is to choose a very high mass M for the corresponding gauge boson; the experimental limit on the proton

lifetime of � 3 0 years implies M > 18 GeV. 10 10 Higher exceptional groups All exceptional groups contain the subgroup G2 , and therefore the colour group SU(3)c . In the following table we display the decompositions of the various ex c ceptional groups into SU(3) x flavour group as well as indicate the transforma tion properties of the basic and adjoint representations. In the previous section we have argued that a possible description of the c strong, electromagnetic and weak interactions is one based on the group SU(3) x SU(3)flavour . Needed for such a description were six quark flavours and eight leptons: c 2 x (3 , 3) + + 2 x (3" , 3) (1,8) + ( 1 , 8 )

223 "' "' l'>

Table 1

Basic representation Adj oint representation and c x ? Group G G :o SU(3) and decomposition under decomposition under subgroup subgroup

c c -c = c c c G SU (3) 7 = 3 3 + 1 14 S + 3 3 2 + +

c c c c = c c c c F, SU(3) x SU(3) = (3 ,3) + c3 ,3) + (l ,8) 52 (8 ,l) + (l ,8) + (3 ,6) (3 ,6) 26 +

- c -c = c c SU (3) SU(3) SU (3) 27 = (3 ,3,1) + (3 ,1,3) 78 (8 ,l,l) + (l ,8,1) 5 x x + + E c - c c + (1 ,3,3) + (l ,1,8) + (3 ,3,3) - (3 c ,3 ,3) +

c c c c = c c c c E SU (3) SU ( 6) 56 = (3 ,6) c3 ,6) + (l ,20) 133 (l ,35) (8 ,l) + (3 ,15) + c3 ,15) 1 x + +

c = c c c c SU (3) x E(6) 248 (l ,78) (8 , l) (3 ,27) (3 ,27) Ea + + + Below we shall discuss the embedding of the SU (3) x SU(3) scheme into the various schemes based on the exceptional groups. We emphasize that all gauge theories based on the exceptional groups are anomaly free. This is easy to see for all exceptional groups except E6 , since those have only real representations . Some of the representations of E6 are complex (e.g. 27 , 351) ; however , also if these representations are involved as fermion representations, the corresponding 1 0 theory is anomaly free ) . The group is the smallest exceptional group wh ich contains the desired F4 : F4 flavour group SU (3) . Since the basic representation 26 contains three quarks and antiquarks and a lepton octet, we need two 26 representations for a minimal reali stic scheme (altogether six quarks and antiquarks , one lepton and one antilepton octet) . Thus we obtain just the fermion content as required

c c 26 + 26 (3 , 3) + (3 , 3) + + antiparticles 1,8 '\, '\, (4.4) (u, d' , b") + (c, s', h") + leptons + antiparticles

As in the G scheme , the unifying interactions in the scheme cause the 2 F4 decay of the proton into leptons in second order of the gauge coupling , thus superheavy bosons are necessary .

The group E6 : Here the basic representation is 27 dimensional. The minimal scheme one can construct is essentially equivalent to the scheme discussed above , F4 if we interpret the direct sum SU(3) of the two SU(3) groups in the flavour group

SU (3) x SU (3) as the gauge group for the weak and electromagnetic interactions. In c c c c c this case the 27 representation decomposes into (l ,8) + (l ,l) + (3 ,3) + (3 ,3 ); {one new neutral lepton [SU(3)-singlet] has to be added to the fermion content of the scheme}. Two 27 representations of E6 provide us with six quarks and one F 4 lepton octet.

The group E7 : The fermion representation is 56 dimensional , the flavour group is SU (6) . We interpret the group SU(3)flavour as the SU (3) subgroup of the flavour c flavour group, in which case the 56 representation decomposes under SU(3) x SU (3) as folllJwS :

1-:e obtain the desired fermion content (six quarks , one lepton octet) plus four neutral lepton states. The remarkable feature of the scheme is that here we need E7 only one irreducible fermion representation. Of course, within this scheme super heavy gauge bosons are required in order to cure the problems of proton decay.

225 The group is the only simple group , for wh ich the basic and the ad Ea : Ea joint representation coincide . Thus it is required that the boson and fermion representation of the gauge theory have the same structure with respect to Ea the internal symme try group, and is therefore especially suited for the formu Za lation of a supei•symmetric theory of quarks and leptons .

In case of the flavour group is The fermion representation decomposes c Ea E5 • under SU (3) E(6) as follows : x

c c 8 7 8 ) ( S , 1) (3 27 ) (3 , 27) 24 + + , +

thus the fermion content of the theory consists of 78 leptons , 27 quark flavours and eight colour octet "quarks ". The latter are the fermion counterparts of the

colour octet gluons . Note that these colour octet quarks are singlets under the

flavour group , i.e. they would not participate in the electromagnetic and weak interactions .

The colour octet quarks as we ll as 23 of the "normal" quark flavours must be relatively heavy : the ob served hadron spectrum can be described in terms of only

four quark flavours .

Since the flavour group is of rank 6, there are many possibilities to as E5 sociate the generators of the electric charge with one of the self-adjoint genera

tors of Consequently there exists a considerable amount of freedom to assign E6 • the electric charges to the various quarks and leptons . In order to be specific,

let us consider a particular scheme for the elec tromagnetic and weak interactions .

We start from the subgroup SU (6) SU (2) of The decomposition of the basic x E5 • representation of under the subgroup SU (3) SU (6) SU (2) is x x Ea c c e e 24S (3 , 6 , 2) + o· , 6 , 2) + (l , 1, 3) + (l , 3 5, 1) c e c (S , 1, 1) +(l , 20 , 2) (3 , 15 , 1) 1 ) + + CJ" , TS,

flavour Let us further assume that the SU (3) subgroup needed for the descrip

tion of the weak and electromagnetic interactions is an SU(3) subgroup of SU (6) . c flavour The 24S representation can be decomposed under SU(3) SU(3) as follows x

c c e e -2 4S 7 . [_ (3 , 3) + (i , 3>] + S (l,�)+14 (l , 1) c c c + (S , 1) + (3 , + (3 , 6) 6) Thus we obtain:

21 quark flavours with the charges 2h , -1;, , -1/g1 flavour 6 quark flavours transforming as the 6 representation of SU (3) . These quarks have unconventional electromagnetic and weak charges -- in particular the charge 4h appears. 226 14 SU (3)-singlet leptons: they are neutral and uncoupled from the "normal" weak interaction.

flavour - 8 SU(3) octets of leptons, among those 32 leptons of charge 1 .

Obviously the Ea scheme is a very complex system, containing about eight times more fermions as the 32 fermions needed for the contemporary phenomenology of hadrons and leptons. At the present time it might look ridiculous to view such a scheme as a realistic one . However , if future experiments should reveal the existence of many new quark flavours and leptons as well as of new interac

tions, the E 8 scheme might have to be regarded as a serious possibility to describe the pattern of leptons and quarks in nature.

5. OUTLOOK

In this lecture I could only discuss a few aspects of a possible understanding of QCD and QFD within a larger scheme of interactions . One of those aspects I tried to emphasize was the proliferation of the number of quarks and leptons , which is a feature of many schemes. How can a physicist who is trying to reduce the number of elementary constituents of nature to a minimal number live with this fact? One way to understand the large number of leptons and quarks would be to interpret them in some sense as composite objects. However at present there exists no evidence for a substructure of the fermions : no excitations of leptons and quarks with higher angular momenta and different parity have been observed . The fact that the anomalous magnetic moments of the electron and muon agree to very high accuracy with the values predicted in QED suggests that the electron and muon have no in ternal structure. If a substructure of quarks and leptons exists, it is presum ably one occurring at a more sophisticated level than the kind of substructures studied in atomic , nuclear and hadron physics. Ano ther , and perhaps more convincing way to live with a large number of "elementary" fermions is to change our attitude towards the physics of leptons and quarks. It could well be, as emphasized by Abdus Salam, that even on the level of the "elementary" constituents nature does the same as anywhere else: ·rt is rich in structure, but based on very few principles . Thus, if we regard principles, e.g. the gauge principle, a particular gauge group , etc . as the "ele mentary" constituents of nature, we need not worry about how many leptons and quarks exist -- be it 32 or 248.

227 REFERENCES AND FOOTNOTES

1) H. Fritzsch and P. Minkowksi, Phys . Letters 63B (1976) 99 .

2) H. Fritzsch, Proc . of the Orbis Sc ientiae 1977 (Coral Gables , Florida) .

3) J. Pa ti and A. Salam, Phys . Rev . D .§. (1973) 1240; Phys . Rev . D 10 (1974) 275 .

4) H. Fritz sch and P. Minkowski, Ann . of Phys ics 93 (1974) 193 ;

5) M. Gell-Mann , P. Ramond and R. Slansky , to be published .

6) Georgi, Quinn and Weinberg, Phys . Rev . Letters 33 451. H. H. s. (1974)

7) H. Georgi and S .L. Glashow, Phys. Rev . Letters 32 (1974) 438.

8) F. Gursey and P. Sikivie, Phys. Rev . Letters � (1976) 775. P. Ramond , Nuclear Phys . BllO (1976) 214 : M. Gunaydin and F. Gur sey-;--:r:- Math. Phys. � (1973) 1651.

9) R.D. Schafer, Introduction to Non-associative Algebras (Academic Press, Inc. , NY , 1966) . N. Jacobson, Exceptional Lie Algebras (1971).

10) H. Georgi and S.L. Glashow, Phys . Rev . D 6 (1972) 429 .

228 APPENDIX

BASIC PROPERTIES OF THE EXCEPTIONAL GROUPS

Group G2 : The group G2 is the group of all automorphisms of the octonion algebra (see, for examp le, Ref . 9). Below we summarize some properties of G2 •

Fundamental representations: 7, 14 (adjoint representation 14) . Rank : 2

7 x 1 7 + 7 + 7 + 14 + 20

The other exceptional groups (F4 ,E5 ,E7,E8) are the automorphism groups

of the Jordan algebras one can construct using the matrices of the type

We mention the following properties of these groups .

Group F4 : Fundamental representations : 26 , 52, 273, 1274 (adjoint representa tion 52) . Rank : 4

26 26 = + 26 52 273 324. x + + + Group E5 : Fundamental representation 27 , 78, 351 , 3511, 2925 (adjoint representation : 78 ). Note : E6 is the only exceptional group which has complex representations , e.g. 27 , 351 Rank : 6

27 27 27 + 351 3511 x + 27 = 1 + 78 + 650 x 27 Group E7 : Fundamental representations : 56 , 133, 912, 1539 , 8645 , 27664, 355750 (adjoint representation: 133). Rank : 7

56 56 1 + 133 1539 1463 x + + 1 1 + 27 + + 27

229 Group Fundamental representations : 248 , 3875, 30 , 380 , 147 , 250 , 2 450 240 , Es : 6 696 000 , 146 325 270, 6 899 079 264 (adj oint representation: 248). Note: Basic and ad joint representation coincide .

Rank : 8

= + + 248 x 248 1 248 + 3875 30 380 + 27 000

SU (2) : 248 = (56 ,2) + (133,1) (1,3) Es ::i E7 x +

b -----· -···-·-'--- b

b � >---<__ ,. ,_____

+ N' µ Vµ µ+ µ I \ I \ I \ I \ I N' \ N" \ j \ I \ • \ I •3 \ E-t ) + E-( + • M • ) M \ I \ I \ N'0 I \ N" \ I I \ 0 I \ I \ I \ I ' I N' e- Ye e- e L R

' b" d' u s c u h" c

\ I I \ I \ I \ \ I I \ I \ I \ I \ I \ I \ I \ I \ I \ I \ I \ \ I I \ I \ I \ I \/ " v v h" ' b" d' s L R

230 MORIOND PROCEEDINGS

N" First Rencontre de Moriond 2 vol (1966)

N" 2 Second Rencpntre de Moriond 2vol ( l967)

N" 3 Third Rencontre de Moriond 2 vol (1968) FF

N" 4 Fourth Rencontre de Moriond vol (1969) FF I N" 5 Fifth Rencontre de Moriond vol ( 1970) I N" 6 Electromagnetic and Weak Interactions (197 1) FF

N" 7 High-Energy Phenomenology ( 197 1) FF

N" 8 Two Body Collisions (1972) FF

N" 9 Multiparticle phenomene and inclusive reactions (1972) FF

N" Electromagnetic and Weak Interactions (1973) FF IO N" 11 The Pomeron (1973) FF

N" 12 High energy hadronic interactions (1974) FF

N" 13 High energy leptonic interactions (1974) FF

N" 14 Phenomenology of hadronic Structure (1975 ) FF

N" 15 Charm , Color and the (1975 ) FF J N" 16 New fields in hadronic physics (1976) FF

N" 17 Weak Interactions and neutrino physics (1976) FF

N" 18 Storage Ring Physics (1976) FF

N" 19 Leptons and Multileptons (1977) FF

N" 20 Deep Scattering and Hadronic Structure (1977) FF

N" 21 Color Symmetry and Quark Confinement (1977) FF

DOES YOUR LIBRARY HAVE

STANDING ORDER ?

R.M.l.E.M. Laboratoire de Physique Theorique ct Particule� Elementaircs Biitirnent 211 - Unive"ite Pari,-Sud ORSAY (France) 91405

19 LEPTONS AND MULTILEPTONS

CONTENTS

I - STORAGE RING PHYSICS

F. Laplanche, ,, Evidence for resonant structure near 1780 Me V in e+e -annihilaiion observed at DC! (Orsay) ,, ; F. M. Renard, « Theoretical comments on the w' (l.78) vector meson ,, ; V. Luth, « KO production in e+e .:...annihilation ,, ;H. K. Nguyen, " Direct evidence for charmed particles in e+e -annihilation at Spear ,, ;M. Perl, « Evidence for, and properties of, the new L. charged heavy lepton ,, ; V. Blobel, « Recent re sults on e+e-annihilation from Pluto at Doris ,, ; W. Wallraff, " New results on•e+e-annihilation at energies around the charm threshold obtained with the Dasp detector at the Desy Storag� Ring Doris ,, ; W. Bartel, « Neutral and· radfative decays of the 111/Jparticle ,, ; M. Greco, " Radiative decays of old and new mesons ,,. ; . S. « Matsuda, « Pair production of charm in e+e-annihilation ,, ; M. Davier, Expectecfphysic · �s at Petra-Pep Energies ,, .

II - DEEP INELASTIC SCATTERING

H. Spitzer, " New electroproduction results from Desy ,, ; C.A. Heusch, " Son:re topicsconcer ning vector meson in charged lepton-nucleon scattering ,, ; J. J. Aubert, " Aciual limitations in the deep inelastic experiment and how the European muon collaboraiion can improve it ,, ; F. Hayot, « Description of inclusive hadron production in deep inelastic lepton-nuc1eon scatte ring ,, .

III - NEUTRINO PHYSICS

C. Baltay, « Dilepton production by neutrinos in neon ,, , P. Mine, ,, Dimuons and trimuons produced by neutrinos and antineutrinos ,, ; C. T. Murphy, " Search forneutrino induced dimuon events in the 15 foot neon bubble chamber with a two plane EM! ,, ; A. Benvenuti, « Properties of neutrino induced trimuon events ,, ;C . Matteuzzi, " interactions in Gargamelle » G. Ber ve ; trand, « Observation of a likely example of the decay of a charmed particle ,, ; A. B. Entenberg, " Elastic neutrino proton and antineutrino proton scattering ,, ; J. C. Vander Velde, " Neutrino interactions in the 15-foot hydrogen bubble chamber search for charmed particles ,, ; L. R. Sulak, " Experimentalstudies of the acoustic detection of particle showers and neutrino physics beyond IO TeV "·

IV - THEORETICAL LECTURES

M. Gourdin, « Currents quarks and high energy experiments ,, ; J. Kaplan, " Some aspects of weak interactions in models with quarks or more " ; H. Fritzsch, " Flavourdynamics of 4 leptons ,, ; M. K. Gaillard, " Asymptotic freedom and deep inelastic scattering ,, ; C. Quigg, " Issues in charmed particle spectroscopy ,, ; L. L. Wang, ,, Anticipating the intermediate boson » ; K. Kang, « Flavour-changing neutral current� in elastic and deep-inelastic neutrino scattering ,, .

V - SUMMARY K. Goulianos, " The finish (summary and conclusion) ,, . 20 DEEP SCATTERING AND HADRONIC STRUCTURE

CONTENTS

I - ELASTIC AND DIFFRACTION SCATTERING J. Lach, « Hadron elastic scattering - An experimental review " ; J. Favier " A charge-exchange exclusive reaction at !SR energies " ; M. le Bellac « Theoretical ideas on the Pomeron ,, ; U.P. Sukhatme « Some new aspects of high energy pp elastic scattering ,, ; B. and F. Schrempp « Probing the space (-time) structure ofhadronic diffraction scattering " ; A. Kernan " Colliding pomerons-experimental situation and outlook ,, ; F. Niebergall " Search for exclusive double pomeron exchange reactions at the !SR " ; G. Goggi « Recent results on exclusive reactions at the !SR "·

II - HIGH ENERGY INTERACTIONS WITH NUCLEI W. Busza « What have we learned from hadron-nucleus collisions about the extent in space and nacure ofhadronic interactions " ; A. Capellaand A. Krzywicki " High energy hadron-nucleus ? collisions " ; L. Caneschi « High energy interactions on nuclei ,, .

III - LARGE TRANSVERSE MOMENTUM PHYSICS M. Jacob " Large transverse momentum phenomena " ; R.D. Field « Implications of the recent large pT jet triggerdata from Fermilab " ; D. Linglin " Observation of large pT jets at the !SR and the problem of parton transverse momentum " ; R. Miiller " Jets and quantum numbers in high pT hadronic reactions at the Cern !SR-Preliminary data- " ; R. Baier " Oppositeside correlations at large pT and the quark-parton model " ; J. Ranft " Comparison of hard scattering models for particle production at large transverse momentum, opposite side rapidity distribution and single particle distributions " ; R.C. Hwa " Nonscaling hard collision model " ; S.J. Brodsky « Large transverse momentum processes and the constituent interchange model ,, ; W. Ochs " Hadron fragmentation at small transverse momentum and the parton model "

IV - MISCELLANEOUS I.M. Dremin " Clusters in multiple production " ; E.J. Squires " Pion-production, unitarity ahd the multi-ladder pomeron " Benecke " Long range correlations in multiple production of ; J. hadrons " ; T. Inami " Enhanced violation of the OZ! rule for production of thec\Jmeson in the central region "; R.C. Hwa " Quark-parton model in t�e fragmentation region " ; W. Ochs « Scaling laws for energy inclusive measurements " ; B. Diu « Search for non-regge terms in two-body amplitudes : some results " ; F. Grard " Observation of narrow peaks at Ge Vin 12 2.6 GeV/c pp interactions " ; M.J. Teper,,tjf /J production in hadronic collisions : a critical review of models ,, ; K. BOckmann " Inclusive production of vector mesons in hadronic interactions ,, .