149 T H I C K - T A R G E T B R E M S S T R A H L U N G 1257

spread of these electrons and positrons is mainly caused by multiple scatterings, and not by the production mechanisms.

Some qualitative features of Iya)(t,k) may be under­

stood in the following way. Since we are interested in k comparable to the incident-electron energy E0, the energy of the electrons E from which these Y'S are produced must also be very close to E0. Now the electron spectrum given by Ie

a)(t,E), Eq. (22) changes its shape abruptly at /=0.75 r.l. For £<0.75, we have IeV(t,Eo)= oo ; whereas for *>0.75, we have Ie<M(t,E0) = 0. This tells us qualitatively that practically all the high-energy y's are produced from t=0 to 2=0.75, and after 2=0.75, the intensity of the Y'S is just attenuated by the absorption factor e~

7/9^~°-7b\ a s s n o w n j n pjg. 3.

I. INTRODUCTION

MANY papers1,2 have been devoted lately to the idea3 that much as in the case of the hydrogen

atom4 a noncompact group G should be used in classi­fying hadrons. All baryons are then expected to lie in one infinite-dimensional unitary representation (i.d.u.r.) DB of G. Of course, the mass of the higher members should increase as a consequence of G-symmetry-breaking effects, but one should be able to foretell unambiguously the spectrum of baryon resonances that follow the lowest lying 56 states. Similarly all mesons are expected to belong to an i.d.u.r. DM of G. Con­cerning G, the following candidates have been dis­cussed1"3 G = *7(6,6), GL(6), and U(6)XU(6)XO(3,l). We wish to show in this paper (Sec. II) that existing

* This work supported in part by the U. S. Atomic Energy Commission.

*Y. Dothan, M. Gell-Mann, and Y. Ne'eman, Phys. Letters 17, 148 (1965); C. Fronsdal, Trieste Report No. IC/65/68 (to be published).

2 A. Salam and J. Strathdee (Trieste Report) have developed techniques for constructing vertex functions for £7(6,6) and GL(6,Q.

3 R. P. Feynman (unpublished). 4 See, e.g., A. Salam, in Proceedings of the Oxford International

Conference on Elementary Particles, 1965 (Rutherford High Energy Laboratory, Harwell, England, 1966), p. 241.

In Figs. 4(a), (b), and (c) the curves for klya)(t,k),

given by Eq. (24) as functions of t and k/E0 are plotted. The computer programs (in ALGOL) for generating all the numerical values given in this article are available upon request.

ACKNOWLEDGMENTS

The authors would like to thank Professor W. K. H. Panofsky for encouragement, Dr. Z. Guiragossian for discussion on some practical aspects of the thick target, and Dr. John Butcher for an advice on numerical analysis. We are also grateful to the authors of many SLAC technical notes on similar subjects, the diversity of these treatments having prompted this investigation.

experimental evidence, especially on the baryon spectrum, is sufficient to exclude all these three possi­bilities. In Sec. I l l we will propose an alternative possibility based on the group U(12)XU(12XSL(2C) which has at least a chance of survival besides some speculatively interesting features.

Let us still recall the differences between the groups mentioned above: SL(6,C) (U(6,6)) starts with a given Z7(6) (U(6)XU(6)) multiplet and keeps adding s-wave mesons to it in the most symmetrical way. Thus at each step the maximum spin increases by one unit and also new SU(3) representations appear. SL(6,C) and £/(6,6) thus lead to large spins paired with large unitary (SU(3)) spins. U(6)XU(6)XO(3,l), on the other hand, starts with a given U(6)XU(6)XO(3) representation and keeps exciting its constituent quarks into higher and higher orbital states. Again, maximal spin increases by one unit upon each step, but no higher SU(3) representations are introduced. High spins appear with low unitary spins. We could label SL(6,C) and £7(6,6) as "SU(3) and spin explosive" and U(6)XU(6)XO(3,l) as "orbital explosive" groups.

Now in order to get a precise idea about the particles appearing in a given i.d.u.r., we shall always write out the few lowest terms in an i.d.u.r. By lowest terms we mean lowest representations of the maximal compact

P H Y S I C A L R E V I E W V O L IT M E 1 4 9 , N U M B E R 4 3 0 S E P T E M B E R 1 9 6 6

Noncompact Groups in Classifying Hadrons*

P. G. O. FREUND

The Enrico Fermi Institute for Nuclear Studies and the Department of Physics, The University of Chicago, Chicago, Illinois

(Received 15 April 1966)

We show that the noncompact groups U(6,6), GL(6,C), and U(6)XU(6)XO(3,l) proposed by many-authors to underlie a classification of hadrons can be excluded on the basis of existing information on the hadron spectrum. A scheme which at least avoids the obvious difficulties of the above noncompact groups is mentioned.

1258 P . G . O . F R E U N D 149

group K of the noncompact group G under consider­ation. For the three cases under consideration, these are K=U(6)XU(6) for G = 17(6,6), K=U((>) for G=GL(6,C), and K=U(6)XO(6)XO(3) for G=U(6) X27(6)XO(3,l). Let us now go through the details of each case.

II. EXPERIMENTAL EVIDENCE AGAINST THE NONCOMPACT SYMMETRY GROUPS 2/(6,6),

G£(6,C), AND U(6)XU(6)X0(3,1)

We start with 27(6,6). The baryons are then classified in

DB= (56,1)++ (126,6)-+ (252,21)++ • • • , (1)

where the parentheses indicate the lowest U(6)XU(6) representations and the signs the corresponding parities. Now from (1) we see that by the time one reaches spin J (which is experimentally observed in the 2-BeV region) one has to tolerate 6104 states, which is a rather large number. Whereas this may be argued away against "further experimental investigation," it is important that quite a few of these 6104 states are expected to be K+p resonances below, say, 2.5 EeV. This seems to be out of the question since it is well known that the K+p total cross-section curve is as good a smooth line without any structure as one can hope for in hadron physics.

We next consider GL(6). Here the argument is much like the one just produced for 27(6,6). We now have

£> i 3=56++700-+4536++

Whereas against £7(6,6), the smaller group GL(6) is slightly more economical with respect to the number of low mass (If < 2.5 BeV) baryon states, it again predicts numerous K+p resonances before even accounting for the already observed resonances.

We now come to the case of £7(6)X<7(6)XO(3,l). When reduced with respect to the maximal compact group, which in this case is Z7(6)X£7(6)XO(3), one uses

Z>*= (56,1,2*0),

where Ro stands for the well-known representation of 0(3,1) that reduces into L = 0 , 1, 2, •••. Now the economy of low-lying states is improved; in fact, as we shall see "too much improved." The point is that no matter what the orbital angular momentum L, the £7(6)X £7(6) representation (56,1) contains only octets and decuplets but never a singlet. Now the famous F0*(1405) has /p=J~~, and it is by now reliably estab­lished that it is an SU(3) singlet. Thus F0*(1405) cannot belong to DB given by (3). Thus one would have to use a second baryon representation to incorporate F0*(1405). If already the next to lowest baryon states require a new infinite dimensional representation, then the whole purpose of noncompact groups would be defeated in this case. We thus conclude that £7(6)

XU(6)XO(3,l) (in spite of the fact that it does not involve K+p resonances) is also excluded.

The question now is raised as to what the features of a successful noncompact symmetry group Gs have to be. On the basis of our above analysis, we can draw the following conclusions:

(a) Gs should contain U(6)XU(6) as a subgroup. (b) In Gs, DB should start with the 56 baryon states

but should not involve any low-lying K+p states. This implies that the next to lowest baryon states should belong to representations contained in the product of three-quark representations [i.e., (56,1), (70,1), (20,1)] but no representations that contain qq pairs.

(c) At least one of the representations (70,1) or (20,1) [more likely (70,1) since it can decay into (56,1) and (6,6*)] should appear in DB in order to accom­modate F0*(1405).

III. AN ALTERNATIVE GROUP

In short, the results of our previous section show that the symmetry should be of the orbital-explosive type but should tolerate alternation of 56, 70, and/or 20 representations. Such cases have been encountered in the case of shell models of baryons.5,6 We shall refer to a very specific example and discuss both the baryon and meson spectrum to be expected. Consider instead of Z7(6)Xf/(6)XO(3,l) the compact version U(6)XU(6) XO(4)~U(6)XU(6)XSU(2)XSU(2). The quarks are here in the representation ( 6 , 1 ; | , | ) ; the baryons are in (56,1; §,§) and the mesons in (6,6; J , | ) . This can be more easily visualized by imagining the quarks to be in the central potential such that 6 s quarks (6 states) are degenerate with 6 p quarks (18 states.) At this point we have enlarged the compact part of the symmetry group but we still have not produced any SU(3) singlet baryons. To achieve this we further enlarge the compact group to U(12)XU(12)XSU(2), which amounts to mixing "half" the orbital angular momentum with SU(6) to £7(12). At this level our aims are all achieved even before extending to non-compact groups. We have the representations listed in Table I.

We see that the baryons up to spin f + 3 = § are in an economic 1456-dimensional representation that starts with a (56, 1, L = 0 ) and follows up with a (70, 1,L=1) that contains F0*(1405). [The U(3) X£7(2)Spin for all representations in the table can be trivially derived from the last column.] The mesons are deserving of a comment. We see that the 36 s- and 108 ^-quark-antiquark states are unified into a single 144-dimensional representation. Thus insofar as the mesons are concerned, our scheme closely resembles U(6)X 17(6)X0(3,1), but we avoid all obvious baryonic troubles. Our scheme essentially amounts to the syn-

5 O. W. Greenberg, Phys. Rev. Letters 13, 598 (1964). 6 P. G. O. Freund and B. W. Lee, Phys. Rev. Letters 13, 592

(1964).

149 N O N C O M P A C T G R O U P S I N C L A S S I F Y I N G H A D R O N S 1259

TABLE I. List of representations.

Particles 17(12) X ££(12) XSU(2) U(6) X ££(6) XO(4) 17(6) X 17(6) XO(3)

Quarks (12,1; J) (6,1; i f ) (6, 1; L = 0, 1) Antiquarks (1,12; i) (1,6; J,J) (1, 6; L = 0, 1)

Baryons (364,1; | ) (56,1; i,}) +(70,1; i,f) (56, 1; L = 0, 1, 2, 3 ) + (70, 1; L=l,2) Mesons (12,12; 0) (6,5; 1,0)+ (6,6; 0,0) (6, 6 ;L = 0, 1)

thesis of hadrons from degenerate s and p quarks. Although so far we offer only a classification of hadrons, at least we are not in violent disagreement with experi­ment. In particular we predict a (70, L=l)~ to be the next baryonic supermultiplet and a (6, 6*, L— 1)+ to be the next mesonic supermultiplet. In terms of a quark model, mass formulas come to mind which essentially amount to the fact that the Gell-Mann-Okubo spacings in all the higher multiplets should closely parallel the basic 56-plet (S*— Fi* should be the same for all higher octets and decuplets as it is for the basic octet and decuplet, etc.). Of course, an orbital explosion can be produced by embedding7 £7(12) XU{\2)XSU{2) in the noncompact group [7(12) X£7(12)X»SX(2,C), but the existing hadron spectrum does not unambiguously warrant such a step. The construction of three-particle vertices for our group is possible and will be dealt with elsewhere.8

7 If we had started from 27(6) instead of U{6) XU(6), we would have had the compact group £7(12) XSU(2) and the embedding would lead to U(12)XSL(2,C). I t is interesting to remark that in this case the compact U(12)XSU(2) is a subgroup of the group of Greenberg's model (Ref. 5) which, as is well-known, follows an SU (24:) classification, again based on equivalence of s- and p-quark states. However, in extending the model of Greenberg to higher shells one enlarges the group at each step and produces very many new states (even after spurious states in the sense of Elliot and Lane are discounted). Our scheme is, in this respect, more economical, since arbitrarily high spins can be reached through the orbital explosion SU(2) —> SL(2,C) without ever having to further increase the group.

8 Another possibility, of course, would be to consider groups generated by infinite-dimensional algebras rather than infinite-dimensional representation of noncompact groups. To our knowledge very little has been done so far along this direction; see, however, S. Okubo and R. Marshak, Phys. Rev. Letters 13, 818 (1964). The incorporation of Regge ideas is also an open avenue. We still wish to mention the interesting possibility that, whereas in the construction of the "ground-state'' hadrons the strong 5 U (6) -symmetric inter-quark forces lead to SU(6)-multiplets, in the construction of excited hadron states the inter-hadron forces with their possibly large violations of SU(6) and

IV. CONCLUSIONS

From our analysis we conclude that the noncompact groups £7(6,6), GL(6,C) and £/(6)X*7(6)XO(3,l) are not compatible with the existing hadron spectrum, but that £7(6) X £7(6) need not be the maximal symmetry of hadrons, since schemes such as the one discussed in Sec. I l l can be constructed which accommodate all the existing hadrons without over-populated supermultiplets.

Note added in proof. Based on the same idea of intro­ducing Z = l quarks along with the basic L = 0 quark, another interesting possibility presents itself besides the group treated in Sec. I I I . As in Ref. 6, one con­siders the group £7(18). One then reduces this according to the chain £7(18) -> U(6)XU(S) - » £7(6)XO(3). The three generators of 0(3) are interpreted as in Ref. 6 as orbital angular momentum. The remaining five generators of £7(3) (second ring in the breakdown chain) are interpreted following R. Gatto, L. Maiani, and G. Preparata (private communication) as quadrupole generators. One then obtains the multiplet structure of Ref. 6 but with the possibility of parity changes within a given multiplet. This scheme would, however, run into trouble that the basic quarks appear in an 18-dimen-sional (L=\) representation and no L = 0 quark sextet can be accommodated in it. This difficulty can be over­come if, much as in Sec. I l l , we consider a group that places L=0 and L=\ quarks in a common 24-dimen-sional representation this time, however, of £7(24). The breakdown chains one could follow would be (a) £7(24)-> £7(18)X £7(6) followed by the above men­tioned breakdown chain of £7(18), or (b) £7(24) -> £7(6) X£7(4) -> £7(6)X£7(3) -> £7(6)XO(3).

even 5£7(3) symmetry could abolish the SU(6) [or even SU(3)2 structure. Consequently the excited hadrons would not appear in complete 517(6) [or even 5£7(3)] multiplets.