Mesonic spectrum derived from the Weinberg sum rules

Nuclear Physics BI32 11978) 155-175 © North-tlolland Publishing Compnay

MESONIC SPECTRUM DERIVED FROM THE H'EINBERG SUM RULES *

Claude LEROY Physics Department. "File Unirersity. Southampton S09 5,Vtl. England Received 28 February 1977 (Revised 29 August 1977)

A saturation of the system of the Wcinbcrg sum rules, uqng the Brout-l'~nglcrt-Truffin saturation scheme, ix perl\~rmed. To include the spin-tqip sum t'ulc'~in the treatment. without blatant contr~Miction with experiment, an association la'.', ix introduced. The algebraic '~chclllC presented here provides u'~ with at duality-tYl~Cspectrum for inesoll~ with m a n y relll;.irkably Sll¢Ct'ssfuI illzlS~,relation,; and Iestable r e l a l i o n s :.|lll,t)ng COlllqing L't)ns|~ln ix.

I.

Introduction

In the problenl o f ck, ssifying elementary i',:uticlcs, the discovery of SU(3) symmetry [I ] has led to scmle remarkatfle systemalics. The mesons m:,y be assigned to the representations I, 8 of lifts group and the baryons to the representations I, 8, I0. The simplest inlcrpret,tlion of this classificatiem is to suppose tile existence ¢)t" quarks 121 classified in tile rcl+resentation 3 of SU(3). It,+tile qtmrk model, tile mesons arc built I'tom qltark-antiquark pairs and the baryons from three quarks. So we obtain a I~hen¢mlemflt~gical law which explains tile classification o f the experimentally observed mesons and baryons. Mesons (baryons) having quantum ,lumbers which du nt~t correspond to the representations I, 8 ( I , 8, 10) o f SU(3) are called exotics. The quark model naturally corresponds to an SU(6) X 0 ( 3 ) symmetry and in Ill;It language, tile mestms may be assigned to tile representations ( I , L), (35, L) whereas tile barytms to (20, L), 156, L), (70, L). ('hiral symmetry SU(2) X SU(2) constitutes ant)tiler al~prt~;lch to tile study of the spectrunl and transitmns amtmg elementary particles. It is generated by vector i and axial currents written in tile quark model as J ,i = q'fla7"i¢I, Ju5 = ¢lT,~tsrtq • The quark model introduces an explicit prescription for the equal-time cotnmutatt~rs

* Work

SUl~portcd IW the I.I.S.N.{l~cleium). 155

156

C: Leroy / Mesonic spectrum

between the charges of these currents [3]. These charges are observable quantities. Their matrix elements between various hadronic states are measured in electromagnetic and weak transitions. Finally, a last approach to tile problem of elementary particle spectroscopy uses the general properties of analyticity of scattering amplitudes [4] and asymptotic behaviour "5 la Regge" [5]. The dynamical hypothesis of duality [6] between resonances and Regge poles allows us to relate different Regge trajectories. Exotic channels appear )lot to contain any resonances; one makes the assumption that tile exchange forces are absent in these channels. This hypothesis simplifies some of duality equations and allows their resolution [71. Tile characteristics of these solutions are: (i) Relations among Regge trajectories, associated with tile absence of exchange forces in tile exotic channels, called exchange degeneracy. (ii) Relations among the residues of these trajectories. A meeting point between the chiral symmetry approach and duality, at least m its local form, (Venezia|lo model [8] ) has been established by Lovelace [0 I. By imposing the Adler consistency condition [I 01 , he has derived relations among masses and decay widths of the mesons coupling to the rrrr system. These results are tile same as those implied by current algebra and chiral symmetry. An important step in the unification of duality and chiral symmetry has been realized by Brout, Englcrt and T1uffill [I I1. They have sht)wn that the saturation of tile sum rules [X '(r), X ~(0)] = 0 (essentially Ctluivaletlt to the Wcinbcrg stttll rule II; see below) is realized either: (i) Via pairwise cancellation of particles, i.e. the con tributitms to tile SUlll rules arc grouped in pairs of particles belonging to different Regge trajectories but with precise relatitms among their masses guaranteeing tile saturation of tile sum rule, or: (ii) Via daughter cancellations i.e. the contributions of a resonance to a sum rule arc cancelled by resonances with the same mass and decreasing spin (daughter rcso-

nallces}. lu this way, they have made the following predictions: a) a mass spectrum, described in terms of linear and parallel trajectories, in remark, able agreement with experiment; b) relations amt)tlg couplillg constants not in contradiction with experiment. The aim of the prescnt pal+er is t() c()ntinue the search t)f more explicit links bctweeu the duality and SU(2) X SU{2) chiral symmetry approaches. We choose to saturate the system of Weinberg sum rules [ 121 which may be written IX ~, X - ] = 2 13 ,

(i)

[x + , IX ~, M 21 I -= O,

(tO

[X ÷, [X ~, M j ~ 11 = o ,

(~0.

In these equations, X is an axial charge operator (X +" ~ X I +- i X 2 ) , / j tile third gonerator of isospin, M the mass operator and J the total spin (J-~ ~ j = _+/./2).

C L eroy / Mesonic spectru,n

157

in this paper we propose to saturate this system o f sum rules by' extending the Brout et al. solutions to the sum rules II1. Starting from the commutator Ill taken between one-particle states {hereafter called external pairs) with helicities differing by' one unit. we saturate with intermediate one-particle states and we apply the procedure o f pairwise cancellations and/or daughter cancellations, in fact. if we naively extend the Brout et al. procedure to the sum rules 111 the rest, Its are catastrophic. We propose a simple and systematic "association law" which will allow us to solve the system o f Weinberg sum rules for the elastic case without modification of the Brout et al. resuhs for the sum rules !I: w6 obtain a minimal spectrum of mesonic resonances which is very similar to that o f the quark model. We do, however, need a finite number of extra low-lying trajectories. The paper is organized as follows. In sect. 2, we summarize the content of the Weinberg sum rules. More extensive discussions can be found in the literature, h| that section, we also review the saturation scheme of Bmut-Englert-Truffin. Sect. 3 is dew~ted to solutions for the saturation of the three Weinberg sum rules when the external states are L = 0 or L = 1 states. The general solution for external slates with arbitrary L is reported in an appendix. hi sect. 4. we compare our results with experiment.

2. The Wcinbcrg sum rules :rod the Ilrout-E.iglcrt-Truffiu ,~lturation scheme 2./. 77tc ff'ci, hcrg s , , t n & ' s These sum rules can bc obtained using I'CAC [131 (partially conserved axial current ), the comnmtators between axial ch:ngcs themselves anti with the generators or' the I)oinc;.ird group, at,d the :.lbsetlce of resommces (and thus of l,?,egge tn,jectories) with isospin 2. The stun titles are:

I.V*, .v I = 2/3,

O)

[;t", [,v', a/"llt :~ = 0 ,

(11)

l.V*, [.v',am~l It ~ = o ,

OIt).

Originally, Weinberg [12] derived tile sum rules I and II as a consequence of tile existence of unst, btracted dispersion relations and tile asymptotic (s -, ~') vanishing of the t-channel amplitudes with isospin 1 :rod 2 for the reactions rr:' + 0~ --- rrh +/3. This depends on the hypotheses about the intercepts of the leading Regge trajectories: o~/~t(0} < 1,0~/_~(O} < 0 . lie obtained the sum rule 111 by writing a superconvergence relation for the isospin-2 exchangc part of the helicity-tlip amplitude in the reaction rra + ot ~ n h +/3. Tile axi:d charge operator appearing in the sum rules (1)-(11I) is related to the tr:msition amplitude M(~ '-,/3 + rr:*) of helicity 7, for the process ~(M p) --./3(X, p) +

158

C Leroy / Mesonic spcctrum

7ra in a collinear frame through

]'.,M(ot ~- ~3 +rr :', M _

7i /]

The following properties of tile operat~lr X will be extensively used:

(a)

(~3. XIX"I~, ,~) = -n,~rTa(- 1 ) G - J 0 (~, -XIX~I~, -X)

for ~ = O. this condition means that

(/LOIX~t~.O)= O

it'~

= 5~

(where ~, is the uatural parity of a meson i o f intrinsic parity ~i and spin Ji and is defined: 5~i = r / i ( - i ) ; i ) (b) We will use in tile followitlg,/-spin reduced matrix elements i.e. (13(I~. M~), ~, I,Vl¢l ailed,

,lla),

,~) = (- ' ,llc~ G i.l/,ll~ t;~ 31 +-,1I~ (~t-, Xll)tllc~, ,~)

where ~lli is tile third c o m p o n e n t of the isospin I i. For brevity we will write

<~, 7~ll,vlle~.

X> ~:

i,~x~

2. 2. H i e llr~ ,it t-I'.tlgh'rt. Tru.l]i. set tura ti~ ,it si'ltonc

hI this r,ar~er, we shall satunat,..: the system tfl weinbcrg sum rules with an infinite spectrutlt. Such an apfm~acll tnakes sense only if we have at our disposal a dynamical mechanism whidt tells us Imw tile contributiorts of tile various particles (belonging to families uf Regge trajectmies) cmnbine to satisfy the sun| rules. Such a mechanisnl is oflmcd by tile llrmn-I(nglert-Truffin l I 11 .,:,aluratJou sdleme at least to satl.,rale tile sum ,ules II. We will see later under what conditions this scheme can be extended to include the sum rules III. lu their work, lhtnlt et al. (we ;ue t\dlowing these autlmrs ,.:h~scly thrtmghuut this paragr:q'dl: see ref. [ 1 11) interpret tile equati,.m l X ' ( o k .V~(r)l = 0 .

(2.1)

as l\dlmvs: from tile ex:.lcl SU(2) X SU(2) symmetry we have dXi/dt = 0 (.V ' being the axial charge). On tile other hand ill ;, ¢lw;uiant no,malisati~m

]'. g.,tAll Ilcuce eq. (2.1) saturated with one-particle states in tile infinite IllOllle,l[tlln ['lallIO is the statemeut that the l'orw:,rd amplitude for r d A - ~ rt-B* vanishes ]br all cm'rg.v. This statement ilnplies th:it every ,nonlent is/.ero at t = O.

159

C [,crt))' /Jlc~,rJntc S p c t ' t r u m

/~dvd'

hn T(v. t = O) = 0 .

Indeed (2.1) implies the vaifishing of the hi~hcr derivatives:

(A,,(r) - -d;.n:- .V+(r) )

-4-

.4-

[.'C(O). A,,lr)] = 0 .

.V,~(r) being replaced by (--~112)n with ~ l l the mass difference between the intermediate state and the "'target". Thus the interpretation given to (2.1) requires that 7-(/.,. t :: O) be infinitely superconvergent (ISA). Brout ct al. find ;.).solution to the etlutHions o l supercot|vergence [I I I v,hcn the intermediate states belong to linear and parallel trajectories. That soltltion implies that the amplitude may vanish for a discrete tmmber ¢)t" values o f t inchMing the Adler p()itlt value which t'llstlrcs the link with chiral symmetry. In order to present the explicit solutions of Brout et al., let us consider the amplitude Y(s. t - O) describil)g the reaction ,"t" A - ~ r r - A + (s-channel) at t = 0 a,d ~vliich ()beys tiu ulistibtractctl dispersioil rclution

T(s,t

I

O)=

/- ~ hn 7"(~,". t

IT •

]

,~"

-

0 )

S

-ds'.

('tit'.; ate trot c.nsidc~ed; their (m|issi.n is cquivalcul t() the neglect ()t" multi-particle couligut;ith).s in Ihc t-chunucl. This hyp()lhc.',is is als,) exlcmlcd h)~' : r o d . ch:m.cls. l.hklcr these assuml'qi()us, Wc will L,sc ll~lllC)V., IC~.()ll~lllce sattl0alion ;.ld crossill,,.~.symII)C[l'x, ' [() cqtklte the Icnidttc ()1' (Co111111o11) 1CM)ll~lllCes IO .~' allLl 11 CII;II|II(2]S. we ol')t.'lill

Z;[

, .tl,,(x),.] =,,

,, l.s (:

MD

u

M;;

•

O,

317t = O ,

s ~'u

")

2,11"x ,

I

p = i(.v

u)),

which lcadn to + ..... n Lu + p .

I. n

2 ~.{X} v. II

1;1~

=0, u_l

=0.

I)-

Since T(p. t) is a mcromc)rphic function of u. this result must bc identically zero cveryv, hczc. T h e . it is nccess:uy that the residues at the poles p = u. vanish for all n. The P.ssiblc solutions ;Ire thcll: dccoUlqm ~ solution ot daughter c;mccllalion solution (it is the strictly local sattu;iti.n); (2.2a)

160

C Leroy / Mesonic spectrum

v,, = 0 '

mass degeneracy solution (3/~ = M~) (it c a n n o t be true for all n) :

(2.2b)

~3,,(,~) = L , ' ( ~ ) . ,

v , = - v , , or 2 M "a

=~..

~1n + M 2 d :

pairwisecanccllation,

i.e. cancellation between it and it' contributions.

(2.2c)

These results are sufficiency c o n d i t i o n s to satisfy tile snna rules II (first m o m e n t ) . Brout el al. have shown that if there exists a solution to tile complete set oFthe sttperconvergence equations, the resulting spectrmu consists of particles belonging to linear and parallel trajectories with definite relations a m o n g their masses as mentioned in the i n t r o d u c t i o n of the present paper. The goal of our paper is to find what are the sttfficiencv conditions For this spectrtuu to lead to a solution of the third sum rule. The right-hand side of sum rules Ill being zero, we are. by necessity, restricted to sufficiency conditions on the Brout el al. spectlunl. Indeed It) st+lye these stuu rules we adopt a strategy similar to the o,lc used for the sum rules I1: as m e n t i o n e d in the introducti~m, we start l'roul tile c o m m u t a t o r III taken between one-particle states with heine|ties differing by one unit and saturate with itltermcdiate one-particle states: we then apply the procedure of p:mwise c a n c e l l a t i , n and/ol danghtcr cz, nccllation as described in (2.2).

3. Exmnples In this section° we consider examples of the applicatitm of the Iht)ut ct :,1. satt,rat i t m s c h c , u c l t ) the Weinbctg sum rules (I) ( l l l ) w h c n t h c c x t c , n a l p a i r s a r e s o m a L = 0 tH' L = I states. (The general solltti~m is reported in an al'~pcttdix). This serves It) show t,s the difficulties arising from naive application t)t" this satlualit)n scheme It) Ihc Xtlln rules III. llowcvcr, we shall i~,opose it solt, titm tt~ the sattnatitm problctn i't)r the third Weinl)crg sutn rule. 3. 1. l'.'xtermd Imirs [p. p J

(An. We have a pairwise cancellation soluti(m between the n and tile At c(mtribt,tions tt~ the sum rules 11, Ill with rite extcrual pairs [p(X = 0). D(X = 0)1 [p(X = In, O(,~ = I)l and Ip(X = I ), p(,'k = 0)]. This sob,nitro leads to the m;,ss relations: " = .-(M;,' ,I I,2 M'xt ~I/~'X, "~ = ,ll~l• ,

" l11;)

,

(M',, = O)

( l l ; J ''t" = I ~ - , 1 ( ;

=0-)

(3.1)

and It) the following relations :t,uong coupling ctnlstants: , .2 ..~V",(O)~, = s)t,;,,(O)

, ~ -2 0 ) . + ~X;,h(O) .- ,~t,l~(

(3.2)

C. Leroy /Mcsonic spcctrum

161

I -2 I -~?'PA, ( I ) = !.V~,I~ ( I }.

(3.3)

" ' " ' ~ Is ~t'" ( I ) - .'i,~H(I) + v.li'~,rr(0) + S)(~,~l(0) + ." pA l . . ..

"x~,,(o) .

- x),.(o) ,/y_M.,, ~'x/~-".L,.,,(l) x~,,(o) - x/Y.L,,(, )x,.,,(o)~. =

-

My

h(.l eC = 0 - + . 1 (; = I - ) is the daughter of tl. Thus. we obtain the final result:

X~,(O) = ~

.VpAI(1) Xph,(O )

2X/~2.VpH(I ) A'ptl(0).

(3.4)

For the olher c o n t r i b u t i o n s o f the recurrences along lhc rr. A r lrajeclorics we rnttst have datrghter cancellation solutions (see the appendix). (B). Duc to p;rrity conservation, co exclusively couples in helicity I to tire ,"rp channel. The co c o n t r i b u t i o n to the sum rule II with the cxtcl nal pair [p(X = I). p(X = l){ is:

(M;,

,U~)l.VL~.# ~)l

-

=

o

(3.s)

leaving us with tire solution ,1/~, = AI L . Ilmvcvcr considering tire co contribt,ti(ul to tire sum rtrlc III with the cxlern,,I pair [p(X = 1 ). p(X = 0)], we arc led to tire dcct~trpling rcsttlt: 1

,v~,~(

{ )

=

o.

This decor,piing is ~dwiot,sly tmaCCCl~lal)lc. The soh,tion, wc propose which avoids this dilfict,lly is an "'associ:flion" t~l"the rr, A j. II (with ils daughter h) c~mllihttlions to the ~ ¢ontJil)t|lior]. The ch,,r;,ctcristics of this m c d u m i s m will bc cxplainctl.al the crrd of this section. Cormcrving, o f ¢otrrsc, tl,c mass rclz, titms (3.1). (3.5). wc then obtain lhe resttll:

x~,.(o)

" l)l.V,,A,(0) ,~""7,~( I } = x/-~ 'IvAI(

X/2 Xvld0)l

(3.6)

(3.6) rcpl:,:ilrg (3.2). As we will scc later, the rclali,m (3.6) is Icslahlc. Wc note also, that lhe ".'tssoci:|lion' used here, le.,ves unchanged the rcst,lts (3.2) and (3.3). Allolher stdtiti(m avoiding tile dec¢>ul~ling Xm.o( 1 ) = 0 would consist in adding new tr:tjectorics. For cx;|mplc, wc ct}uld irrtroducc a "co'" (I ~; = I - , j l , = 1 - , M~, M~). This lr:tjcclory is expcrimcnt;,lly t,nobserved. I!ither co' is J I'c'= 1 - and co' becomes t, ndistinguish;,hlc {'rorrl p or co' i s J t'c' = I - + , but it is an exotic of second kind frolll Ihe point of view of the qt,ark model. One could think to introduce a trajectory j r ' = I ~ with isospin I and degenerate in mass wilh the p tr:rjectory bt, t such a choice wot, ld lead to a parity doubling experimentally t, nobserved. For the A2 c o n t r i b u t i o n , we have a daughter cancellation solutioll. Taking into account the ctmstraints imposed by sum rules II and III, we have the resuh:

!V" {1)=,'~.,,(I) 2" pA 2 ,

(a2(J I'C

=

1 "-)

(3.7)

is the first datrghlcr of A,). 1,1 terms o l ' w i d t h . (3.7), used with the

162

C I.cror /.lh's¢,mc spt'ctrum

experimental result [14] I'(A 2 ~ p r r ) = 71 McV predicts l'(a 2 ~ prr) ~ 177 M e V .

of course, for the m o m e n t , there is no experimental evidence for the a 2 which is the isospin I = 0 partner o f the p' (first dat,ghter o f f). (C). We must also consider the c o n t r i b u t i o n s of the 6 trajectory. Parity conservatiun implies the decor, piing ,',t'~,(0) = 0 and more generally Xpc,,~u)tO) = 0 (where 0¢~,(j) is any m e m b e r of the ,5-trajectory). In fact. we have ollly daughter cancellation solutions for tile c o n t r i b u t i o n s of that tr:.,jectory. (D). We have also to consider the c o n t r i b u t i o n s of ;.111.v-trajectory (.v. a,"c = I + I (; = 0 - , M x = MI~ = 2.5 M~). See fig. 1 ) for which we have only daughter cancellations, x is consideled here as tile / = 0 partner of the first recuHence of r~'. More generally, with tile exterual pairs [c~o(j). e¢,o(j) ] c%(j) = 3 - - . 5 - - , .... pairwise c;,ncellati-ns are possible between members of the 6 trajectory and ot" tile x trajectory: to see this. iu detail, wc i-el-er the reader Io tim appendix. We h~,ve seen that both puirwisc and daughler-type canccllalitms are in difficulty when we naively apply them to the sum rules II1. Indeed we are led to tile utlacceptable result: N,,t..,( I ) = 0. "1"o avoid these difl'iculties we propose a solution which ¢unsists of an association mechz, nisul defined as folh~,,vs: the decoupling A'AI,(X ) = 0. deduced tlOtll tile sum rllle III with the external p;.lits IA,AI is avoided through an ass,ciatiou of the conlribttlitms of pa, rticlc B with particle (' with Mc = MA. This associatiott law solves the ploblcm of the extension to the st,m rules III of the I~rout el al. satttr:ltiotl schetne. %Vc will sec in :1 second eXamlplC ;.tll~)lhet ;q~plication of this asst)ciation l,tw. .x_. l::vl('r,al pairs [.,I 2. A 2 ]

(A). We have a l',ahwisc caucellalion solulMn between the p and g ctmtril:,uthm,; t~ the sutu rt,le II with Ihe exterual pair [Az()~ = I)...12(h = I)1. This solulitm leads t~ lhc mass relation:

(aq,: aq,)= and [o the I'elatiotlalIIoXlB

c .x) cotil'dhigCOllSt:llllS:

!V 2,\2.o(I)=-~V: i .2 ." ." Azg (1)- "V -~ A21.'(I)+ :,.l(A:,o"(l).

(3. t))

The dat,ghter ¢al~cellation solution for the g contributions to the sum rule II with the exte,nal pair I,-i :(X = 2), ,I:(X = 2) I gives: !.,(2 2" A

I-,~ = X ~ ,.r , ( 2 )

2/.', ' - ~ '

'

(3.10)

If we insert the :.lbove rest, hs into sum rules III with the e×tern,d pairs [,'12(~ = I).,12(~. = O)l. I..12l,~ = 2L,-12(~ = I)1 we obtain: .VAzj,(I) = 0 ,

(3.1 I)

C Leroy/3h'sonic spectrum

163

/ N / ~ ,I(A2g(-) ,ItA2g(i ) -- - XA2 f'(2) XA2f'(I ) = 0 .

(.,.I 2)

Concerning the f contributions to the sum rules II, Ill with the external pairs [.'l,(X = I),A:,(X = IJl, [Az(X = 2).Az(X = 2)], [,-1,(~, = I),A2(X = 0)1 and [A : (X = 2), A +(X = 1)], daughter cancellations irriply the following restllts: 31~x~, = Mp ,

(3.13)

- A T x , r ( I ) + ~XA2d(I ) - - 0 ,

(Y14)

XA2f{2 ) = 2 X A 2 f ( I ) .

(3.15)

Ouce again, the result (3.1 I ) is unacceptable. To avoid it, we will "associate" the contributions of p, g, f (and their daughters) conserving the mass relations (3.8), (3.13). Eqs. ( 3 3 ) ) - ( 3 . 1 2 ) and (3.13)+ (3.15) are then replaced by g - . A2f(2) + AA2~,(I) = --- "+ ~

2XA2 r{ 2)XA2 f(I ) + ~

,*'/A2g(- ) ,~ A2g{ 1 )

XA2t,'(2) XAzf'( 1 ).

This second example of "association" displays tile essential pruperty of this type of solution: ]11+o2 = M t+,,,) 2 ' 71/+'. = kl~I That is to say when we want to avoid the • #% 2 decoupling ,X'AidX) = 0 imposed by sum rules Ill with external pairs Ia,al, we associate Ihe co,,lrihutions of particle C with M c = M/,, to tile particle B cont'ribillions. ((' accompanied with its daughters aml (or) particles aheady connected to it through a ca,.io.'llalion mechanistn). The alternative to the associatitm procedtue would im,'ulve itltroducing new tiajechuies. In tile present case, we would illtit,dli++:e either a I ' tiajecttiry with M], /II~, t,, a 3 + trajcct,.,ty with ]~I~, +M~.. +l'hese parity dotnblitigs being experimentally tmobserved we do not retain this Stdlitio,l. (B). We have also to coitsider the ctmlribulitms of the r/:ind rl' trajectories. I'aiiwise cancellations between the ColltiibtltiOllS t.ltll(]') of the rl Ii:,]ectory :Hid the ¢Olllribtiliolls Ciu,(#") of the r/' Irajeclory ca,I oc¢l.,r wh¢,i ]. f ~ -i. Wheli J. f ~> -1 we ,,Itisl h:lv¢ dzilighler C;.lllcell,ilio,is. . . . .

3. 3. I'/wernal pairs [B. B ] The ¢u, n-A,, atlt[ <5 trajectories can ctmtribtite in this case" for the ¢,.J and rr A, trajectories, we have pairwise cancellations between the A z :rod the A, - I I cmltlibt,tions and between the ~ and the 2,~- -- 2~. * contributions. The latter cancell:ltion leads Io the resuh: ]1,1,+u

"x'"'><+ t ) = )t~ .x'~,.,,(0). [ f w h:ls no d:,ughter, the sum rule II for the external pair [B(X .-- 0). B(X = 0)]

(3. I~,)

164

C.l.eroy / Mesonic spectrum

gives Xu~(O) = 0 (compatible with recent experiments [I 51 ) but via (3.16): ,li'tl~ ( 1 ) = 0 .

(3.17)

If c~ has a daughter (~f), (3.16) means Xu~o(I ) < XBw(O).

(3.18)

The results (3.17) and (3.18) are clearly incompatible with present experimental data [15]. So here, the difficulty met in extending the Brout eta[. saturation scheme to the sum rule Ill is not a decoupling but rather an inequality iqcompatible with experimental data. 11owever. we have enough freedom in our saturation scheme to avoid these difficulties. We can "associate" the w and ,5 contributions. This leads to kl13 X n i ~ ( 1 ) ,lt'l~ i ~ ( O )

i'w

(1~- - is the first rectnrence of fi) with" the nlass relallon'/1[[I'~ __/l/,52 = - (Jilt" - /1[/~)" and the detinitions Pi.j ~ ](2 M[i 317 Mi ), v i =- ~ (MT# - 31; ). This solution is somewhat inconvenient in that the "association" used, does not follow our previously announced systematic law. Indeed the "legal association" of co and x contributions (M x = llltl) would lead, ifx has an isosrfill 0, to the relation -

A

'

I

"

. . . .

'*

-l

M,.,,

) + a ;,.,.(t ) = 7Wl,

-~

I

~

"P

•

•

•

x,<.o(o),

i.e.again either to relation (3.17) or (3.18). I f x has ;.ill istispill l, we would el)lain the sohil ion:

.-M,;

-'

with the results (fully colnpatible with experiluents [151) that: Xll,.o( 1 ) 4- 0 ,

Xllw(O) = 0 ,

Xn,.o( I ) > X m o ( O ) ,

if cur does not exist if c~ t. e x i s t s .

I l o w e v e r such a s o l u t i o n i n w ~ l v i n g x w i t h an isospin I su p p o se s the e x i s t e n c e o f

two j#'c = I ~ - multiplets in addition to the B-multiplet at the level L = 1 ": the x-nmltiplet itself and the I n-nlultq~let+ " where I +,{ is the first rectirrence of the 17'. The pairwise cancellation occurring between the A2 and the A I -!1 contributioris does riot present such difficulties. It leads to the mass relation:

-- -(,wfl

)

* A spectrum involving two multiplets I t- - (/. = t ) in addition t~ tile B-nudtiplet has bccn obtained f o l h ~ mg other :ll~pr~aches; see ref. [ 16 I.

C Leroy / Mesonic spectrum

165

and tile final relation among coupling constants: X2

~ ~[A BAI(I)_,I{21.1(I) =N/-, ~ (~v/6XBA2(1)XBA2(0)--~ XBa2(1)XBa2(0)}

For the other contributions of tile recurrences along the w and rr-A~ trajectories, we must have daughter cancellations. (see appendix). 3.4. Sum ndcs I We conclude this section with tile consideration o f the sum rules I and we show which constraints may be derived from them. The right-hand side of these sum rules is non zero. tlence the saturation scheme adopted to saturate tile sum rules II, II! is of course not applicable: for every external pair, we have ao infinity of contribt,tions to these sum rt, lcs without any dynaluical mechanisnl to control them. Ilowever, these sum rules present an interest when we have experimental information abot, t some decay channels. They then allow us to put botlnds on other decay widths. For example, the sum rule I with the external pair (p(X = O),p(X = 0)) leads to I'(A I --, ,on: X = O) ~< 241 MeV: with tire external pair (p(X = I ), p(?, = I), it gives F(A a ~ pn; X = I) ~ 214 MeV. From tile suln rule I with tile external pair (q, r/) we find a bound on F(w r ~ rrr/). Indeed, using tile expcri,uental results [I 4] I'(,5 --,.r/rr)~ 30 MeV, I'(Az ~ r/rr)~ 15 MeV, we obtain I'(¢of ~ fir/)~< 118 McV. Rec:,ll that w t is a scalar particle with isospin I and negative (;-parity. Thus the rr r/decay channel ought to be tire rnost important I'~r wf. If our esti|natc of tire width makes sense, this :trgues strongly against the existence of col. "I'o snzn,narize this section, we can say that both pairwise and daughter-type c,mccllati~m arc in difficulty when we n:nively apply them to tire sum rules III. Indeed we arc led to un:tcccptable results such as Xpt,.,(l ) = O, XAzp(I ) = O. To avoid these difficulties, wc propose a solution which consists in :m association mechanistn defined as follows: the decoupling XAIj(~ ) = 0 deduced from tile sum rules III, with the external pairs [A, A] is avoided through an association of the contributions ~)1"particle B with particle (" with M c = M A. This association law solves the problem of the exlensioll to the sum rules II! of the Broul et al. saturation scheme.

4. Comp:lrison with experiment

In this section, we compare our results, obtained by saturating the system of Weinbcrg sum rules for tile elastic cases, with experiment. 4.1. Quark mo¢h'l Our spectrum o f the " p a r e n t " trajectories (see fig. I) obtained by saturating tile system o f Weinberg sum rules is identical to that predicted by the quark model

166

C. Lero.r / M e s o n i c s p e c t r u m

,E.

f

J

rr

rl

f

rf l

2

J

l'ig. I. The mu,amk" spectrum obtainc.d by ~;;,turating |h,: W~.'inh~.,r,.: ';urn rulc'~ (I) .t k l 2 (tirlil,~: .4.11~).

M2

111|): J rcrsu~

supplcnlcnted with a limiled Imnlber of'additional (muliiplets) trajectories. These iJle :

(a) For evely even L(L > O) j#'c" = / ,

"

jl'(." = L-

~

(b) For even odd 1, jI'C = L ~ -

As an example, we refer to the table I below I'~>rL = O, 1,2, 3. Concerning daughter trajectories, we have systematically exchided tlmse with (.iuall[l.lln numbers incompatible with the quark model.

4. 2. Mclosh analysis (a) Now we ¢
C. Leroy / Mcsouic st?ectru,t

167

FJblc I Sl',,..'ctruln f r o m the saturation ,.chcmc

Quark model s = 0

0 -+

0 -+

s=l

I--

I--

s = 0

1*-

1+-

s=l

2++.1 ++ . 0 ++ . . -~++ . i*+ 0 + + + 1 + -

s =0

2 --+

s= I

,--.,--.,--,--.,--.,--.'--

s =0

3+-

3+-

s = 1

4 ~+..~'++..'++

4 ++.~'+*. 2 ' ÷ + 3 + -

L=O

L=I

2 -+

L=2 2-"

I.=3

t i o n s : l m o n g tilt: d e c a y z m q ~ l i t u d c s L = I -" L = 0 + rr: ( A l -- prr) I

( A : -" prr) t = 2 ( B -" w r r ) L ,

V'~2( A , -" 7yr)o

(5 -~ Tl~h, = (B -" ~ h ,

(4.1)

•

(4.2)

T h e r e s u l t s 2?t~,A i( I ) = .'t ~,ll( I ). ),'llt,.,(O) = 0 ( w t t l o c s m~t e x i s t ) , d e r i v e d f r o m t h e s,ttltt:llion scheme,

;.ltld i t l t t o d u c c d

itllo ( 4 . 1 ) , ( 4 . 2 ) a l l o w us to l',rctlict t h e f o l l o w -

m g d c c : l y w i d t h s ( s e e t~tl~lc 2).

'l'ablc 2 I'rcdiction (in McV)

I':xpcrin)cnt (in McV)

I ' ( A 2 - , ptr; ,k = I )

71

71 ( i n p u t )

l ' ( l l - , t.~tr; h = I )

80

80 -150

I'(AI ~ ptr:X = l)

122

?

I'(ll~pn;h

183

?

= I)

I ' ( A 2 - , r~n'; h = O)

15

I ' ( A I ~ pn; • = 0)

104

I'(~, ~

,Tn: h =

O)

27

15 ( i n p u t ) ? I"6

~ 50

C Lerov I Mesonic spectrum

168

W'e o b t a i n as w e l l a ratio 1/31 = ,l(#~ a l ( I ) l X p . % ( 0 ) l = 1 . 0 8 " We note that i n d e p e n d e n t l y of the Meh)sh paramotrisation our scheme alh~ws us to predict For example: F(o -+ nrr) = 675 M e V ,

( e x p e r i m e n t : 4 0 0 - 7 0 0 MeV)

I'tp' -+ rrrr) > 162 MoV. (b) To prevent the decouplitlg , I t ' l l ( I ) = 0. (see sect. 3) we have associated the n, A t , tl, h, m contributiolis to Weinberg Stlln rules with extertlal pairs [p, Pl. As a resuh, we obtained

.t'~,~( t ) - .\'~>.(0)= =

"

~"

~ I

.t'oa,( I )[.t'~,,.~t(0)+ -- ~,t'~.lO) "

+

2'*Pht

rtVr52 X~tl(0)l

,

(rl = ~i).

#"

l:rom these relations, we extrac't: X~(I

) = 0.60,

(4.4)

which corresptlnds to r/= +1, I'exp( p ~ lrrr) = 145 McV aud to l'(h --> np) = 400 MeV. (4.4) is collipatible with (i) the value Xpw( 1 ) = 0.75 -+ 0.15 c o n l p u t c d by GelI-Mann, Sluirp, Wagner froln the dcc:iy to ~ n' + 7 arid (ii) the b o u n d Xow(I ) ~< 0.62 o b t a i n e d fronl the sum rule I with the external p,lir [p(Tt = I ) , p ( X

=

1)1.

5. COlichlshlll We have saturated the system of Weiilbcig suin rules (iilcludiilg the thhd oile) liSilig lh¢ S:.llUr:llion scheme of I h o i i l t.,I al. hi oldel Ill c×Iclid Iheir sl.'helli¢ Io the third Siiln Itlle w i l h o u i blatant couiradiclion.~ w i l l i experilneill, it proves liOi:cssaiy 1o illlrodtlce :in associaliOli law. Si ringont tests o f lids associatitlli law are difl'iculi 1o filid because iliOSl o f the l:Otil)lilig COliStalitS iilvolved :ire so far experilnelll:illy ill-detorllliiietl. Ncverllieless, the ctluplilig consiaill ,t(mj(l )obtained froin till :l~SOcialioll w:.is shown Io be lit agreclnelil w i l h the value obtaiiled by (;oll-lMalUl cl al. l'rolll the dee,i)' 6o - it + 7. Altogether die algobrai¢ schcllle pl¢StHited here provides ilS with a "dualit),-tYlle" speclruln Ior the nlesoliS with liiaiiy reill:irk:ibly sticcessl'ul mass relations and lestable relations allltlll~ coupling t~t)llSt;.lillS.

* May bc compared wilh: 0.67 '~: 131 "~ 1.60 (rcf. I1%1 ) 13i =0.48 ± 0.13 (rcf. i l 9 b [ ) II/I = 0.89 -+0.07 (rcf. l l9cl). 0.0(~

C. Leroy / Mesonic spectrum

169

l would like to thank I'rofs. Robert Brout and Jacques Wcyers for their help throughout the course of this work. Thanks too to Dr. Anthony tley for useful discussions and careful reading of tile mant,script; to Mrs. Jan Jones for her care in typing the manuscript.

Appendix The appendix is devoted to the description o1 the solutions we find saturating the Weinberg sum rules ( I ) - ( [ I I ) when the external pairs arc t\mued with L states of s p i n / d e n o t e d by eqq) and beh)nging to "parent" trajectories oq. A more complete discussion about these cases and the solutions when eq is a daughter trajectory can be found in ref. [21], Now just two points of notation. When we will have a pairwise cancellation between n and n' contributions, we will write {ttlVm' } . Then !ttlym' } must be undeastood as (n and the particles of mass sqt, ared 3/~ which arc on its daughters trajectories pp n' and tile particles of mass sqt, arcd M~. which are on its daughters trajectories}. A notation 1)1"the type {%q)l)t)a),q)} ASS (c~,q')/q)Otd(/')} will mean that the contributions of the spin-/members 0taq). eh(/') of the trajectories e:, and cq,. accomp:,nied with tlmir eventual dat,ghlers are lek, ted by a pairwise c;mcellation solution. The same for % q ' ) and 0qlq'). We then realize the association of all these conlributim~s I\)llowiug tile law defined ill sect. 3.

A.I. I:;vtcrnall,air~ (.I..I): .Ic%," (;a = +(ex: p(I- - ). g(3- ) (a) ~ A I tra/cctorv ( a . I ) / , / ' < 2 J: p:iirwise cancellations: to:1

,

-at

q)+

Ii M,,,_...~t(/)

t

.

. pp

%-Atq)

~M

t

o

t~'rr - A I

t:l

%-AiV

....

)+%-A,(/')} t

o

M t:l

-~M i

(~n-Al

(j)

part icularly

{TTpp I t

(~n - A I

( Z / - . 1)

+ OtI = O

n-

A I

(2J-

(:,.2) j , / ' ~: L/: daughter cancellations. (b) oa trajcct,,rv ( b . I ) j . j' < 2J: pairwise cancellations:

totta(j)l,p o ~ (1"')} ,

1)}

'

,

(...=M ] }

t:o

.... -z?dy

°ttr_Al(J

}

'

C: Leroy , .Ih,~'omc spectrum

17 0

particularly

Sum rules Ill with the external pairs [,l(X = 1 ),,/1,~ = 011 constraint the decouplings X / w ( I ) = O. unacceptable at least when J = p. In that case. to avoid the difficulty, W e USe ;.In association J { uo Pl: a w ( _"~

-- I ) : A s s : a p p

Q~:'r[ : -!. X l

(

2./

-

l ) + e J ' rt- 0A I ( 2 J -

I)1".

(b.21 When J P = J P (P l%r paritvL considerina sum rules II in the maximal he]icity state, '0,e obtain 'Ilgw(J) = M,~. (b.3)j. i' >~ ZI: daughter cancellations. (el 6 trajc('to O' ( c . l ) I)arity conservati,,m t\nbids ,5 itself to o m t r i l ) u t e . ( c . 2 ) / < ~ 2J - 3 , j ' <~ 2J 2: pairv, ise cancellations:

+:(', ,Ile,~ (j) - M i .

M . x ( i , ) ~ M i, .

,v~,,~ (n(X = I/I) = o . R e m a r k : Then" At trajectmy ptcsentstlnel~flh~wiugchar',,cteristic:the mass degeneracy o f II and A t obtained flora the n, .'\n. II, It contnibuti,ms tc, the sum rules II v,'ith extcrnn;41 pails [p(A = J ), p(,'~ = J )[. that implies that in every site Lfl"

line tE:qect~uy there are 2 palticles v,'ith negative (;-p:uily (,ue witln iS-Slfin I, tint other with isospixl 0). ,,I.2. l ! x t c r m d p a i r ~

(.I, J): Jeer,.,.,: ( ; j :

(ex: A , ( 2 ~ ~ ))

p, 71, It,, rl', L': traject~)ries ~,m omtribute. ( a ) p t r a j c c h wt' (a. i ) j . I' "-¢.Z/. pairwise cancelk, tions:

{%~(j) p p (~p(]")} , M%,(i) -,1.,/I ,

2 31~ = .1,

M..(/)-~

+ M;,.

M/' .

particularly : p p p 0~,(2 .l

I )) ,

2 315 : M;,, + ,~,1~../ ! .

Sum rules Ill with the external pairs [J(X = ] ). J(,'k = 0)[ constr:fint the dccmq'flings X / ( I ) = O, t, naccpet:,tfle at least when J = A 2. In that case, to avtfid the difficulty we use an associati,m (ppp%~ ( Z I -- l )} Ass ~a#(.I I' = ]1")}.

C. Leroy / Mesonic spectrum (a.2) W h c n j ~

17 1

= j e (t' for p a r i t y ) c o n s i d e r i n g sum rules Ii in the maximal helicity

,~1),

s t a t e , w e o b t a i n ~T/~ (~) :

( a . 3 ) j . / ~> 2./: daughter cancellations. (b) r / - B , r / ' - E tra/cch~ries (b. I )j. j' < 2J 1: pairwisc cancellations:

r~(i)

Mc,,7,(f ) = Me, e(/' ) -.~1/' ,

~ ~,~(i)

particularly

M; = ,~t;7 + ,~17.

'i rl pp (x,~, (]") + cgl (j ')} ,

:,~: 1 (/) + ~,~=o(/)m' ,7'}, ( b . 2 ) / , / ' ~ 2.1 -~ I: daughter cancellations. l~cmarks: (i) In ever}' site of the r/.B trajectory, wc find 3 particles: one which is a r/ icct, rrcnce and has a positive (;-parity. A second one which is a D recurrence and has z. positive (;-t~arity. Finally, a third one which is an x-recurrence having a negative (;-palily and so cannot c m l t r i b u t c to the rr.l chatmcl. (ii) In every site of the rl'- l! trajectory, we find 2 particles: one which is a 77' recurrcucc with a positive (;-p:lrity. A second one which is an F, recurrence with a positive (;-pality t~t). The rl' 1'i traiectory is degenerated with the ~5 trajectory (,'~1~{ h/~ scc~cct. 3 ) b u t the mctnbcts of the ,5 t r a j e c t o r y h : l v e a n e g a t i v c ( ' ; - p a x i t y alld

~()

c:lnllt)l COllllil)tltc It) 7TJ.

,,1.,¢. I:~vtcr,al pairs (J../): Jt~Y,r ,\l" (;J = (cx: A l(I ~ ~ )), (.I) p trafl'ch~rv ( a . I ) / , j ' < ~ + 1: p,drwise canccllatitm:

_, Mj = i M,,~,(i) -~ M i ,

(a._)/,/

~2J+

,v},,

Mt~.(i' ) ~ ~I/' . I' d:tughtcrcancellaliems.

(b) r/ - B, r/' L'~trafl,ctories ( b . l ) j , j ' < ~ : p:drwisc c:mccllati,ms:

MJ, a ! (i)

= , t I c~,~, , : o(j)

~ 31 i

,

M

,/(j,)

=

M i.(i, )

= Mf

•

(b.2)i, j ' ~. . . ." I.• d:lughter c:mcellalions. Note th,lt parity ct)nservution forbids r7, to con:ribute. For tile same reason, tl, r/' rccturcnccs c:mn~)l ¢tmttibute to the rrJ channel ill helicity zero.

17 2

C. Leroy / Mesonic spectrum

A.4. External pairs (J, J): Jea n_ B; G j = +(ex: r/(0- +), B( 1+ - ) rr, A i , w, 6 trajectories can contribute. (a)

r r - A t trajectories ( a . l ) . / ' < 2J + - , / < 2J + 1: pairwise cancellations: w,

f

:'

~atotl) pp

6rl = 1 ,.'~ + I=0 -.'~ rt_Aiq / O~rr_Al(4 ' ]j',

Maw(/)~3[ i ,

M

1=1 .... °t t r _ AL 11 )

=M

"~ " = ~' j [ 2 , , ~J[~ ~'l/; +

1=o O . , ) - M j , .

n-A

I

( a . 2 ) / i > ~ + 2,/' >i 2J + I: daughter cancellations. (a.3) Sum rules II, Ill with external pairs [J(.?~ = O),J(?~ = 0} l, [.I(X = I),J(~ = I)1. [J(X = 1), J(X = 0)] lead to

X/2

.~3,~(~)

,tl~

=

Xj,.(O)

(*).

So we have 2 possible solutions: (A) cut (¢0 daughter) exists. Then 1") tile;ins:

,rj ,..( I ) < ,rj ,.(0). (B) for does not e x i s t . (*) lllealiS: ,tt'j~(0)

= 0 ,

Xjco(1)

= 0 .

See sect. 3 for the discussion of the c:lsc J = B (b) 6 trajectory (b. I)]. I' < 2J: D, irwise canccll;,tiotls:

2 MJ = M: + M}.

q),. /t/,y,~(/) -~ M i

,

3I~,~(/') -

M i' .

( b . 2 ) j , j ' ~ Z/: daughter ,:ancellations.

,1.5. I.:wvrnal pair.~ (J, Jr): Jcc¢~" (;j = - ( e x : 6(0 ~ ~')). P, O+ B, r / ' I! trajectories can c o n t r i b u t e . (a) 1"/ -B trafect~rv ( a . l ) ] , ] ' '< 2J + 2: pairwise cancellations: (¢x/-:J (/ :- O) does not e x i s t ) ,

C. Leroy / .th'.wmic ~pc'ctrum

~,

17 3

: M ,:0..

-?,Ic.

( a . 2 ) / . / " > / ~ + 2" daughter cancellations. (b) r / ' - E trajcct~ ,rv ( b . I ) j , j' < ~ + 1: pairwise cancellations:

;c%'(/)+cq.:q)ppan'q')+al.q')

},

(al.: (j = 0) does not exist)

2 ~t; ---~ti + ?,t?, Mo~n,(i ) = Mal.:(/) -= M~ .

M~,,(i,)

= M,:,I (f) =- 111/' .

( b . 2 ) / . / ' ~ ~ + 1: daughter cancellations. ( b . 3 ) / ~ J: sum rules II provide us with the solution ?,/~ = ?,/~ . (c) p trafl'ct+,rv: daughter cancellations l+or all the members contributions of+this trajectory; we obtain the lbllowing decouplings ,Tj~(/) (X) = 0 with ~ = I/I, / <~J. ,.I.6. I::vternal pairs (J, J): Jcotn'_ I.> (,J = +(ex: r / ' ( 0 + ) w , rr--A I, fi trajectories can contribute. (a) w, t r - A I tra/cctorics (a.I)/,/' < ZI + 3: pairwise cancellations:

{c~oa(i.)l~p

l. I I %_A,(i')+~._A,(/')t. o

.?'t;' =?,t, + ?'t,,.

.

/11 l I (., = ?'[ I o ~ M/' , ¢~n-AI / ) ,a' n _AI(I '')

~,

Meow(i) :~ M i

•

( a . l . I ) If wf does not exist: Xj,~(0)

=

0,

Xj~( I )

=

0.

(a.l.2) I f w f exists: x/-J

M,~

( a . 2 ) / , / ' ~> 2J + 3: daughter cancellations. (b) ,5 trajectory ( b . l ) j , j ' < 2J -~ !: pairwise cancell:,tions:

{ ~ (/)m, ,,~ q')}, ..'1,1%(j)

-= M / ,

2 Mj : ?,t~ + ?,i,,-',, ,4,1%(j ,) -- ?,i/' .

(b.2)/, j ' i> 2J + l : daughter cancellations. ( b . 3 ) j = J : Sum rules II provide us with the solution: ?,/~ = Mj.

C Leroy / Mesonic spectrum

174

A. 7. L:rternal pairs (J, J): J e a x ; G j = - ( e x : x( 1÷ - )). p. r / - B , r / ' - E trajectories can c o n t r i b u t e . (a) r / - B trail,trot 3, (a.1)/', j ' ~< ?L/: pairwise cancellations:

(Ot~ = I(/-) + (x/=O(/.) pp CtT1 1: ,(/3 + ~ = o (i,)) ,

ill

t=t.

. = M

(xr/ t/J

t:0..-Mi

(Xrt

'

(/~

M

(at= t(/= 0) does not exist)

t : t .... =M ~=o . . . .

~r/

(./ ~

(xO t,1 J

~ Mi'

.

(a.2)] > 2J: daughter cancellations. (a.3)i =J: Sttnl rules !i give the solution

/ll~,,~(i) = M.~. (b) 17'-E trajcct,,rv

(b.I)j, j' ~< ~

I: pairwise cancellatitms:

{%'q) + og,:q) t't~ % ' q ' )

M~, .(,:) = M~,,,:(j> (b.2)j > ~

M i .

+

~l,:q')},

m~,,/(i,)

(cg,:(/= O) does not exist),

-.

M.v:~, ) - Mj,.

• I: daughtcrc.mcellations.

(c) p tra/cctorv: daughter cancellations leading to d c o u t p l i n g s

Xj, a(g)(7~) = 0 ,

~, = I/I, / < J .

References [ 1 [ M. (;eli-Mann and Y. Ne'eman, The eightfold way (Benjandn, New York); D. Spciser and J. Tarski, J. Math. Phys. 4 (1963) 588. 121 M. (;ell-Mann, I'hys. Lctt. 8 (19641 214; Phys. Roy. 125 t'19621 1067; (;. Zweig. CERN prcpriuts: 8182/'1"11 41)1 1'19641; 8419/'1"11 412 (1964). [31 M. (;eli-Mann, Physics I (19641 63; Phys. Roy. 125 (19621 1067. 141 G.F. Chew. S. Frautschi and S. Mandclstam, Phys. Roy. 126 1'19621 1202; G.[:. Chew and S. Frautschi. Phys. Roy. 123 t1961) 1478. [ 51 T. Rcggc, N ttt~vt) ('imcntt~ 14 (1959) 951 ; 18 (1960) 947; P.D.Ii. (_'tdlinx and EJ. Squires, Springer Tracts in Modern Physics 45 119681 1. 161 R. Dolcn, I). Ilorn anti C. Schmid, Phys. Roy, 166 (19681 1768; II. I larari, I'hys. Rcv. l.ctl. 20 (1968) 1395; P.G.O. Frcund, Phys. Rcv. Lett. 2(1(19681 1235.

C. Lero)' / Mesonic spectrum

175

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