Chiral Symmetry Breaking and Quarks (Current and Constituent) II<Sup>*</Sup>

Chiral symmetry breaking and quarks (current and constituent) 11*

P. Stnhaand V P Gadtam

Indian A smr/tation for the Culitvatiou of Science, Jmlavpnr, (_W<;?/7Y«-700032 {Received 25 March 1977)

Tho distinction boLwoon coustiUieiii and mirrcnl. (jiiarks and its effect on the (3, 3 ) 0 (3, 3) model of chiral synimeti y bri'.akinf^ is investigated. This considoratiou provides a resolution to the exist­

ing difficulties faced by GoJl-Maiin-Oakcs-Kenner (GIVIOR) model, Aidiilo the picvious good icsiilts, v liicli could be obtained from GMOR model, are also recovered. Two postulates are absi.racti'd from ((iiai’k model for independent detminmatiou of the Tkdl-Maun constant’ , r, from the two different sources, (i) pscmdoscalar meson piasscs and (ii) meson-imcleon u'-tcj'ins and baryon masses, ^^itllonl identifying the current quarks with the constituenf. ones. Tlu^ f.wo ev^aluations of c (--^ —0-90), show a remarkable consisicncy, uliicJi is a good evidence in favour of the (3, 3) 0 (3, 3) model. In this context, examining tho difference between AT/(3)s|r,ro„(, and Sf^^(^)curn‘,ii a new insight into GMOR scheme is Huggt‘.Kte(l,

A

significant dis­

tinction bef.AVcen S( ^C^)sironff and SU(ti)currcTd f^^af. tlu' former is approximately invariant in the vacuum, while tho laf.ter is not 3Tie chiral symmetiy liinif. of the baryon mas.scis is shovn to be slightly h^ss than tlic nucleon mass and this providt's the chiral invariant, hadronii; mass scahi. The chiial symmetry breaking bai yonic and mcaonie matrix elements, M'hen combined, yield unified fnesou-baryon sum rules, connecting the masses of the pseudoscalar mesons viih those of the baryous. These sum rules ar^^ satisfied within 2% An extension to include the efleet. of isospin breaking is also eonsideicd.

1. Introduction

The choice of (3, 3) 0 (3, 3) representation of chiral *S7/(3) @) <ST/(3) syniinetry breaking Hamiltonian, is believed to be well-justificd on different physical grounds (Gell-Mann 1962, 1964, (Rasliow I'l- Weinberg 1968, Gell-Mann e( at 1968 Daahon 1969, 1971, 1972) However, the parametrisation of the (3, 3 ) 0 ( 3 , 3) model by Gell-Mann, Oakes tKr Remrei (GMOR) (1968), has recently been ques­

tioned by many authors (Scadron I't Jones 1974, Firelis 1970a, MoNarnee and Seadrorr 1975, Prasad 1974, Jones Scadron 1975, Grrnion H at 1976, Sin ha 6:

Gaiitam 1974, 1976) for different reasons First, tho distirrction beiiMien consti­

tuent and current quarks (Gell-Mann 1972, Hey 1976) was not incorporatml in

* A proliminiiry rnport nl’ wcnlc was prebunti'ct ui (ht' 111 Higti En«'r<^v 3'liysifh RympoBium. Bhubaneswar November 197fi,

10 P. Sinha and V. P. Gautam

GMOR modol (Fuchs 1976a, McNamcc & Scadron 1975, Gunion et al 1976, Sinha Gautam 1976). Second, GMOR solution for “ Gell-Mann constant” c, does not agree w'*ith the meson-meson and the meson-baryon (r-terms that can he extjacted i'rom the experiment (Dashen 1972, Scadron & Jones 1974, McNamee Scadron 1976, Cheng 1976) Thud, some constraints on the SU{^) ^ SU{S) symmetry breaking parameters, rule out' GMOR solution for c (Prasad 1974) Foi these reasons, many attempts have recently been made to determine the value of c Irom different soui ces The various approaches include, phenomeno­

logical S(dution lor c (Scadron & Jones 1974), SU(H) ^ ISU(S) generalisation of Goldbcrger-Trennan relation (Jones &. Scadron 1976), consideration of the dis­

tinction bet,V ecu coiLstituent and current quarks (Fuchs 1976a, Sinha & Gautam 1976), use of the baryon chiral syinmctry breakimj {GSB) ]uuanieters (Gunion ct al 1976) etc. lii the phenomenological approach of Scatlron & Jones (197^), a snilue of c (»^ — 1-U) has been found, which agrees with different' o'-term estinla- t'jons, but the distinction betwiicn constituent and current quarks was not consi­

dered, Besides this, the n -N <7-tcrni situation was not, M^ell-scttlcd at that time.

llovvev''oi, aft(M‘ the availability o f the acfairate n-N jihase shift data of Carter al (197J) all the subsequent dispersion relation calculations of nN o'-term (see t-ablo 1) basically agree. At, the same time, the K N cr-torm estimation of von- Hippel Sl Kim (1970) is not considered to be much contioversial Therefore, besides the pscudoscalar meson masses, the nN and the I\N cr-tcTUi values may also provide an independent determination o f the Gell-Mann constant c, if tin;

baryonic matrix elements o f the scalar densities aie correctly known The jmipose of the present wmrk is to further expand our previous analysis I’or mesons (Sinlia Gautam 1976) and to extend our approach to mcludii the CSB baiyonic inati IX ekunents and also the effect of isospni breaking. W(^ shall unify the solutions for GeJl-Mann (sonstant,, obtained separately liom CSB foi mesons and baryons, without identifying the current quarks vith the coiistitumit ones.

Tliis leads to a unified meson-baryon sum rule connecting the masses o f tlui pseudoscalar mesons with those of the baryons This sum ruU^ is very well- satisfied anil expresses the internal consistency of the present approach In othm- rccenf, Avorks (e.g. Gunion et al 1976), involving a new solution for Gell- Mann constant, it has been suggested to reject GMOR solution, w^hilo the present work signilicantly differs from the oarliiu ones for, wo have obtained a now solu­

tion for Gell-Mann constant, but ha\m siimultanoously retained the older GMOR solution, AAdth a now physical interpretation ascribed to if .

The conceptual difference betAveen constituent and current quarks, arises fioiii till' distinction between the /S77(6)w groups of the classilication of the hadronic states and o f the algebraic structure of the Aveak and electromagnetic hadronic currents The first step in this direction, was due to Buccclla et al (1970). HoAvever, the physical signilicance of the phenomenological mixing operator introduced by Buccclla el al (1970), Avas not fully understood until,

following the^ suggestion of Gell-Mann (1972) and the work of Foldy ^ Wont liny- sen (1950), Molosh (1974) had constructed a unitary transformation, connecting

the generators of current Melosh tiansformalKm

is found to wwk very well (Hey et al 1973, Gillman et al 1974) and cxicuisions and applications of this transformation are also considered by many authors (hocHay (1976), for a complete list o f references) In the ideal Melosh t ransforma­

tion exact CVe for all the eight *Sf?/(3) generators has been assumed. This assnumtion has been relaxed by Fuchs (1973) to got a more realistic form of tlu' transformation from current t-o constituent quark basis. In fact, it is still jiossiblc^

to have aai exactly conserved f^U{^)-^?,stron9 ^ unitary transformation, wdiich reduces to Molosh transformat itm in the limit of c^xact SU{'^) for enrrenti quarks (Fuchs 1973) The splitting betAveen ^^U{[])strono '^'C^(3)ciincHf be (examined and the two SV{3) algebras coincide only m their respective isosiiin SU{2) sub-algebras. Tliorefoi-e, the distinction betw'con the two /S'f/(3) algebras is a consequoiKH^ of SH(3) breaking and the two ^Sfl7(3)algebras become identnyil in the limit of i^xact OVC for all the eight /Sl/(3) generators.

As noted earlier (Fuchs 1973, Fritzsch et al 1974) the scalars, Wi’s, trans­

forming as niemlKU'H of (3, 3) © ( 3 , 3) representation of *S'C/(3) (g) i^U{‘<^)currcnt, not havi’! simple transformation propert.ies under /ST^(3) ® ^U{\\)strpno GMOlt model, the OSB Hamiltonian contains two t,erms, one singlet (vt^) and one octet.

(7tg) under ^^U{^)curren^^ Now, each of and transforms as a linear combina­

tion of a singlet and an octet term under BV{ii)strono> that phenomenologically the tot.al CSB Hamiltonian may be t.akon as a combination ol one singlet and one octet term undc]- BU{"^)^trong the relative weight parameter associated wnth the ^^^^(3) octet term, should mi be same for the two diffvrvni repicsentations ol the CBB Hamiltoniaii, under <S'Z7(3)cMrrcTi«

This point is very much important, and wuth proper attention to this distinction, an attempt, has been made to clarify the difference betwi'cn BV{‘.^)stroug and SU{'^)cvrrent-

TIu‘. plan of the paper is as folloAVS. In section 2, the paramotrisation of the (3, 3) © (3, 3) model from the pseudoscalar meson masses is considiu’cd.

Section 2, also includes a brief account of GMOR rqiproach for the sake o f com­

pleteness. The difference betiveeu SU{3)strone ^^i^)current^ examined through th(‘ CBB baryonic matrix eloments in section 3, W'^herc an independent solution for- Gell-Mann constant is jilso obtained in consistency with that obtained in section 2. Applications of the presently developed phenomenology arc made in section 4, to dciive a few sum rules in the PCAC approximation, fn section 5, wo have extended the considerations of section 2, to include the effect of isospm breaking in the (3. 3) © ( 3 , 3) mode]. Lastly, section 6, contains general dis­

cussions and conclusions regarding the new features of chiral symmetry break­

ing, expressed in the present work.

J2

2. PSEUDOSCALAE MeSON MaSKES

P. Sinha and V. P. Gautam

starularfl (li, .*}) 0 (.‘j 3) ropi Moniatioii o f chiral ^T/(3) (g) S1(J{2) Hymme l.i'v Hamillouijiu is (Ccll-Mami 19(>2, J904),

ir

^----

\-cUf,

^{■■■ (}1)

wluMo Mit‘ Mcahirs, an? foriiUHl by bilinear current quark fields,

q -

( i )

⁽²⁾

Obvdnnsly, the “ ( jell-Maim eonslaiil” r. is a direct measure ol the brcakiJig[-of

*ST7(3)c«jrf'«o ralh(‘]‘ than S(J{‘2)i,ttor,o im|)lieit assumption that s<^)cs \iith (•((. (i), is that, like SUmstronO’ liJvakmg o f SU{3)cnrrent

(mlianeed. Tlu' Gell-Mann constant, c, is related to the bare ciirient quark nias.ses through th(^ following relation,

1 JUirl ma—2nis

■v/2 (3)

Anollau' rundaniontal ^VZ7(3)(g) /S77(3) synimotiy breaking parameter* is dcfini*d as,

1 <0|'a'r/-hdd-25.s|0:

^/2 -0|iZM+5d-hs.s‘ |()> (4)*

3\> dotoiniine t.hese parameter's liorn the pseudoscalar meson masses, one needs make t.he usual J*CAC asmuiptiou and consider the vacuum to one meson mat.rix elenient.s of the axial vetitor divergences.

^ {doU-\~cdQii)Ik-\-

J

^c<01 Vy|Pg > i5i8, ... (5) whore — < 0 1 Cf | P i>

To solve for v,from (5), with i — n, K , it is necessary to know the ratio hoi* this GMOR considered the throe point function,

and took

-h«/r(0

^ < P l{p )

1 I

Rk{p')> (6a)

(6b) a(0) and /?(()) being determined by Gell-Mann-Okubo mass formula. But, such an assumption would have been valid only if the UfS formed a nonet under

13

This procedure leads to (Gell-Mann H al 1908) /,, ' HO that

'fti iiml ^ lUi,

( ' ' G M O i i ^ \ /2 -r^ -o -i ' - -1-25 (00

In view ol the distiuetjon between iSU{H)siro„o and nSU(3)cnr7ent^ il' appears that the Holutions (Ce), obtained throii|Th (Ga) and (Gb), cannot b(‘ correctly recognised as the /S0(3)c'urrf'nt breaking parameter, c. However, .solution (Gc) may bi' iiitei-- preted from a different phenomenology, udiich ivill be discussed latei.

Now, the identification of w.^’s a.s a nomd under jSU(3)sironff^ be avoided either by the use of solt meson technique or by transibrming tlu^ p.sendoscalnrs

’/'f's to con.stitueut quark basis, by the applieatioii of Melo,sh 11 ansformation.

The pseudoscalar meson .states in /r’/ s may be reduced by soil nu'son approach to yield,

_ a , ( V2-l-c)(V2+c-).

0 C K iV ^ 2 -}c){V 2 -y ) ’ ... (7) being the mea.suro ol soft ]iion/kaon PCAO correction (Scadron tV Jones 1974, Sinha. & Gaiitam 197G). Tn ordei to solve for c and c' Irom (7), we resort to tho siiggo.stive power of the quark model. The vacuum (expectation values ol the bilinear quark fiedds are individually divergent However, the latio ot such vacuum ex])ectation values may be d(‘t(‘imined with siiitabh^ ]iy saturating the vacuum matrix elements like <0|w?/|0> , with single quaik intermediate states, which is nothing more than tlu^ .standard pole dominance assumption, it. can be .showm from tho properties ol th(‘ Dirac spiiims that

<0|'ai^|0> _ ^u^'u

< 0 | s . s '[ 6 > “ Zsnig ■ («)

The rcnormalisatioii constants Z^’s are exactly unity for a tree quark mod(d.

While, for an interacting quaik model the differ from unity, but., smee, in a quark-gluon theory, the quark fields are renormalised by the emission and absorption of virtual 8TJ{^3)-symmdric gluons, the ratio ZujZg. can be taken to bo unity. Therefore, from (3), (4) and (8) it follows that*

r/r^ c. ... (9)

Eq. (9) is a result of quark field theory and doc,s not involve any identification of the eurrent quarks with the constituent ones Now, w(^ propose our Jirftl postulate, that eq. (9) is approximztely true in the real world of broken chiral

symmetry.

^ Kq, (9) can also be obtained ui a different manner (Gunion et al 1970b),

14 P. Sinha and V. P. Gautam

"I’iio ra,1,io UjriJtK is no louftor unity, as in GMOTl, but, R J Rk {'s/cx.„w„)l{y,/aKmK).

f^’or Ilf) appi’ociablo PC AC correction (a,^ a/c 1), or even for a weaker assump­

tion of equal PCAC corrections for pions and kaons ^ ocr ^ 1), wo bavt) Irom (7) and (9) (Siuba k Gautam 1974, 1976),

/9 -0 -9 8 ,

wliicli moans

'»'>h ^ 2fKWK 7 0

(10)

(lOa)

wliero 7ii - - l{nha-\-yhd)‘ Rfdation (10a) li,a,s also been obtained by Sazdjiaii &

tStern (1975) (see also Sazdjian 1975), m PCAC approximation, considering f ^Sf?/(3) /Sf^(3)]L algebra (subscript L stands for light plane charges) Tliese authors have also demonstrated the validity of (10a) for soft gluon model.

The ratio R J Rk could also be determined without the use of PCAC, by find­

ing the SU{^i)strono transformation properties o f the pseudoscalars Vid with the help of Melosh transformation, as done by Fuchs (1976a). This yields

fKjn.K^

! 2m 12 U % -f so that.

msjm 6*9 or c —0*94.

( H )

(12) A compaiison o f this result of Fuchs, with ours, throws light on the applicability of PCAC assumption The PCAC assumption moans that {i — 1, . . . ,8) can be used as approximate interpolating fields for the psoudoscalar meson octet.

However, while studying the behaviour o f the .ff^’s, Fuchs (1976a) has carefully avoided PCAC assumption and his result is very close to that o f ours. This agreement may be taken as a good support, in favour o f the use of PCAC assumption

3. Meson-Nuolkon SiaMA Teemsand Baeyon Masses

As discussed earlier, the correct transformation properties o f the current scalars m's under SU{'^)„trona play a major role in determining their baryonic matrix element,s from the mass formula. Since the meson baryon tr-terms are rchited to such matrix elements, we shall now considor the modified Melosh transforma-tion, Adth broken 8U{3) (Fuchs 1973). Such a transformation has the general form for i = 0, 3, 8

V~'^UiV — m

J

d^xq{x)V?(iq{x), (13)

TJie parameter, m serves to deteriniue the scale of the matrix elements and LI involves linear combination of the Gell-Mann matrices and Ay. Following Fuchs (1973), we take IJ of the fonii,

U ^ («Ao+6V2Af,)(3/2)i, (14)

where a and h are just numbers when we are restricted to the baiyonic matiix elements of When exact CJVC for all the eight >V17(3) genei atojs is assumed, the quark mass matrix becomes ^’ (7(3) symmetric and as such, U does not involve any Ag term. Then, obviously i^CJ{"^)stron9 ^^{•^hurrent identical. There­

fore, the departure o f the ratio bja from zero is a measure of the difference between SU{S)sirong

Now H' can be transformed to constituent ijiiark basis as follows*

J rf^a:ry(a;)|(ii-|-V2d>)Ay-l-(m~f.&-l-V^fo)AgJ(/(.r) witli

,v — {ca — ('h-\-'^2h)l{a-\ \/2ch).

(15) (15a) Witi transfoim under SU(S)stiong as the indices suggest. The new paiameter. ,s', is quite diffeuuit from tlie Gell-Mann constant c, the luttei being a measure ot the breaking of A S V 7 ( 3 ) c t t r r e n « > ^ b i l y for 5/« — 0, one can have 6* --- c. Since, in GMOH approach, w/s were ticated as a nOiiot under SU{'3)strong it appears that GMOK solution should coriespond to ,s lather tlian to c. Now, it is (wident that the baryonic matrix elements of m t (not cu ) for i 7I 0, are to be determined from the *S7/(3) mass sjilittmgs of the baryons.

The baiyonic exjiectation value of the tiital strong Hamiltonian is**

CBI I B > - 2Mb^ - 2iIio='+ ^ B \H'\B : (

10

⁾

so thiit tlio ^Sf/(3) baryon multqilets satisfy the familiar quadratic mass formulae*** In the chiral symmetry limit the baryons ictluce to the common mass M^^, which provides the chiral invariant principal hadronic mass scale.

In quark-gluon theory, this can arise from quark self-interactions or quark-gluon mtcractioiis Now, according to (15) and (10) it is easy to have,

< NlsitJgl W > == (17a)

♦ Tho analogous oxpreaaioTis givoii 111uq, (3(i) of FuuLh(1973) aic' not correct bocauao tho last two equations in (35) of Fuchs (1973) lu-o not consistent with his paianiotnsaliuii ot U, Wo ai‘0 thankful to J3, Hagchi foi calling our attention to this,

—)■ —>

** Our states are noriualisod co-variantly. < //| p '> -- (27r)‘^27j!i<J(j/~p),

*** llceontly it has boeu oraphosisoil by Fritzsch et nl (1974). that both baryons and mesons satisfy quadiatiu »Vf7(3) mass formulae, The quadratic mass lormulao have tho advantage that they can bo universally valid for tho Iteggo rccuiTonces of both baryons and mesons,

If) P. Sinha and V. P. Gautam

... (I7b) by Ui(^ tiHfi tlui quadialic baryon mass formula.

To (Ictioniuuo llio })aiyonio matrix olomont.s ol Wq, let us fij'st go back to tJi(i iiaiv-(i (|ua»k JuodeJ, vvliere no disimction between current and constituent (jiiaiks IS Tnfid(‘. Now, Zwoig rule allows us a sjinj)le determination the o f the r.il.io****

iN \u,,\N ^ N I imd-dr/+s.s IN >

< iV|iZ7/4'dd—25s|iV > V2. (18)

Siiu (‘ Ike nucleons are supposed to bo made up o f only non-strango quarks; this lelatioji is consistent with tlie naive quark model. However, strictly spealiing, a iiiieleon is composed of only three uon-strange covstiluent quarks, but is a c\)m- pljoaled mixture ol an infinite nnmboi o f strange and non-strange current quarks (hitelis 1078). Thus, -r | .s,s-1 iV > may not be negligible compared to < JV | ww I-r/d|A^ 'riierefore, tlie distinction between constituont and current cpiarks, suggests tliati (IS) would have becMi acceptable, if the transformed as a nonet inidcj’ Nf/(8),sOort(7 No, we should, rather expect,

N\ 1^ ; ... (19)

wlu‘i’(‘ the u’f’s are as defiiuMl in (15). The validity of (19), is oui second poshdate.

Now, iuiy linear combiiiatioii of u.^ and Wy ean be transformed to a linear coriibinatioii of ?e„, and iCy uith the help o f (18) and tlioii baiyonic mati ix elements cun thou be easily determined from (17) and (19),

111 view of the validity of (19), it is worthwhile to examine, how relation (18) IS now affected By the use o f (13), (14) and (19) it can bo easily shown that

V2^ + * = V(I -f" 0 2.

so that wc are again back to (18) It is quite interesting to observe that relation (18) IS not affected by tl^^ distiinction between constituent and current quarks.

Bq. (18) corresponds to vanishing o f < N ] U ' \ N > m the chiral >Sff/(2) g ) S!U{2) symmetry limit (Gell-Mann 1969), like the pion mass i.e.

Jim < X I Uq-^cu^ I X > ' ^ 0 , X = 7t,N . (20)

Sph rpf, a ol Cheng (1976).

17

The ttN and KN <r-t-orm can now, provide an evaluation of c and bja, which will subaoquently determine *■ and Mq. The nN and KN cr-torniB, when traiiB- formod to constituent quark basis, look like

S ~2Mn ^ I a+^/2cb “^a-h V2C6 ' N > (21a)

1 ( V2- i c )

3 2M~~ < N V2(q - lb) y/^a^h)

a-\-y^2cb ® ~ 2 ( a + V 2 c 6 ) “ 2{a-\-y/2rh)

After a little simplification we have from (19) and (21) M i + - -

2 L V V2H-C _ (T” ^^

y/2~\c

N \w ,\N >-\

V3 < N \ w ^ \ N > y

N >

.. (21b)

(

22

⁾

As mentioned in section 1, the recent well-settled situation for nN cr-torm is displayed in table 1. It may be noted that a value of nN a-term, ^ 68 MeV, agrees with all the results quoted in table 1 Wo shall use this value of nN cr-term, together with the KN a-tenn of von-Hippel & Kim (1970), to doter- Ta-ble 1 Results of dispersion relation calculations oi nN (T-terni based on

recent nN phase shift data

Authoi H nN (T-iorni (MeV)

Niolson and Oadoa (1974) 66+ 9 Hito ami .lacob (1974) 09+12

Langbion (1975) 61 + 16

Chao et al (1975) 57+12

Moir et al (1976) 74+ 7

67+ 8

mine c and 6/a. Now, it is easy to have from (17) and (22) c -0-96,

so that from (21),

b j a —0-4.

(23a) (23b) The above solution for c has an excellent matching with the solutions (10) or (12), obtained from pseudoscalar meson masses in the piovious section and the consistency pattern is, therefore, maintained. In order tc appreciate the differ­

ence between SU{2)strono ^^^H*^)cnrrent us calculate the AS^/(3)fl«fonp break­

ing parameter s. From (15a) and (23), it follows that

5 ;= ;-1 -2 4 . (24)

The close agreement of the value of s with GMOR solution (6c), is very much suggestive, because it has been assumed in GMOR scheme that the scalars uts have SU{?i)iirono transformation properties. This observation throws a now

3

18 P. Sinha and V. P. Gautam ^

light on the GMOK solution for tho Goll-Mann constant, and it will be discussed in detail later

The contribution to nucleon masH duo to CSB, | | iV^> , may now bo easily calculated. Thus, the chiral symmetry limit o f tho baryon masses turns out to bo,

if„P S 900MeV. ... (25)

Mq fixes the chiral invariant hadronic mass scale and contributes to most of the baryon mass. Thus, we have <cB\H '\B> < B \ Hq\B'>, consistent with tho concejit o f an approximate chiral symmetry o f tho world.

4. Sum Rulesin the poao Approximation

In this section wo derive a few sum rules in the PCAC approximation, i e., by assuming tho phenomenological PCAC correction parameters a i’s, introduced in section 2, to be unity. However, the sum rules derived in sub-sections 4.1 and 4.2 aio valid oven for the weaker assumption— equal PCAC corrections for the psoudos(^alar mesons.

4.1 Meson and baryon masses

The evaluation o f g from the two independent, sources, (i) psoudoscalar meson masses and (li) nioson-iiuclpon (r-terms and baryon masses, makes it possible to relate the above cpiautities in a consistent manner. According to (10), we have

-v/2+ c

\/2—ic (26)

so that, it IS easy to eliminate c from (22) and (26) Thus, Ave get tho following simple sum rule, connecting the masses o f the pseudoscalar mesons with those of the baryons,

niK 3 (27)

by making use of Gell-Mann-Okubo mass formula. With, 68 MoV and 170 MoV, this unified moson-baryon sum rule is satisfied within 2%. However, if we use GMOR relation.

\/2-(-c V2- i c

instead of (26), avcgot from (22) tho following relation,

m,K^ 3 i f J V ® I trKtK^NNl ’

... (28)

19

which is badly violated. The well-know^n difficulty with GMOR scheme is once more apparent from (28).

4.2 rj-meson.

Regarding the applicability of i;-PCAC, GMOR have argued that, one may expect it to be on a similar footing as kaon PCAC, at least as long as 7j-rj' mixing is neglected. The identification o f current quarks with the consituonl ones, leads to an approximately SU{3)cufrent invariant vacuum and as a consequence of the octet enhanced SU{3) breaking structure in H', rse of ly-PCAC in GMOR scheme, yields Gell-Mann-Okubo mass formula, subject to tlie condition,

fn ^ Jk ^ ftr

However, according to the present analysis, vacuum is not invariant under SU{3)current ^nd as such, by the ust^ of ;;-PCAC one arrives al; a different sum rule*,

¥7r‘“W'7r“ ... (29)

foi’ no ?;-?/' mixing. Tt is true that we have no experimental information regarding the decay constant But, for a valid 7y-PCAC and an approximate >S/?7(3) symmetry, the naive guess is that would be of the same order as fj{. The above sum rule (29), predicts

1 -2 6 /, Si 0-77

in oonsisten(!y with ihe normal expectation Tt must be noted that eq. (29) is not a substitute for Goll-Mann-Okubo mass iormula, which holds true inde­

pendently, as will be seen in section 6 4.3 f^cMlar K-me/fon

The delicate question, whether scalar k dominance of the divergences of the strange vector currents may be considered on the same footing as ordinary PCAC, is discussed earlier by Dashen (1971). Ho has remarked that, if scalar X is a Goldstone boson, SU{3) no longer remains the symmetry group of vacuum, so that this situation may be troublesome for simultaneously having ^f/(3) multiplet o f particles. But, with the new concept o f two different SV{3) groups, strong and current, this is no longer a problem. The existence of SU{3) multi­

plet of particles requires approximate SU(3)atrong invariance of vacuum, while the breaking o f SU{3)cnrrent infhe vacuum is consistent with a picture o f scalar X-meson as Goldstone boson, like, n, K,7j, so that there may not be any lack of symmetry between vector and axial vector divergences. In fact, the present scheme expresses a largo SU[3)eurrent breaking in the vacuum by virtue o f our

♦ Bq, (29) is tho modified form of our previous sum rulo (Sinha &. Gautam 1974). in the derivation o f which, a term <7h^|0> was dropped, hut with <0|wa|0> finite, tho term

<^/l'yolO> IS non-zero,

20 P. Sinha and V. P. Gautam

first posiuJttte, but this doos not moan that the same is true for SU(3)gtronp An ajjproximato estimation of the vacuum breaking of jSU(3)Btrongi based on ti-ansformatioa (13) reveals that it is indeed small enough, in agreement with what is known phenomonologicallJ^ However, one may still think that the mass of the scalar k meson is not small enough to be considered as a Golds-one boson in the ;ST/(3) ®/S'r/(3) symmetry limit. But there is nothing wrong with using an effective k pole to parametrise the strange vector divergences.

(Dashen 1971) Thus, wo can write (DeAlfaro et al 1973)

f M , <0|it,|0> , (30)

which expresses that the mass of the scalar K-raeson, is a direct measui’o of-the

>^^H^)cniri’nt non-invarianeo of the vacuum. Considering similar equations for pions and kaons, it is easy to have by the use o f (9),

. (31) This relation has also been obtained by Sazdjian I'k Stern (1976) in the PCAC approximation, considering | »SV/(3) (g) .ST/(3)]f, algebra If is 1250 MeV (particle Data Group 1976), f , turns out to be, / „ s 0-4/„ . This enahles us to calculate, the 7f,, foim factor by the use o f Glashow-Weinborg (1968) formula

/h. ( 0 ) = ( /kH A = - /« » ) /2 /* /,^ 0 .9 7 , ... (32) which is reasonably consistent with Aderaollo-Gutto (1964) theorem. This is a good tost o f the large SU(3)cvrre„i breaking o f vacuum and also of the sum rule (31).

5. Effect of Isospin Beeaking

We eonsider a more general form o f OSB Hamiltonian, taking into account the effect of isospin breaking,

H ^ UQ-{'CUg^-\-du^.

Tins can be taken as a model o f a symmetry breaking Hamiltonian, which con­

tains some o f the electromagnetic contribution (Gatto 1970). The necessity ol the inclusion o f a a, term is suggested as a solution to both Dashen’s paradox (Dashen 1969) and , 3jr puzzlo (Sutherland 1966). The SU{2) breaking para- meter d is related to the current quark masses by the relation

d = -^-1

^ ~1~ (34)

to order to determine d, we proceed in a manner similar to that adopted to tletermming c in section 2. It is easy to obtain,

( V2-J C + -V * d) { V 2 - I C + - i p ' ) p 2 - l c - ^ p ) ( v2- I c ' - ' ^ - d ' j 7C+'"' K+

Ci5)

where d' — < 0 1 10>/<01 w-„ 10>

Wo, now, make uae oi a poRtulate similar to (0) that d' =■-- d, as may bo abstracted from the (juark model and obtain.

^ ( W2—ir;) - - - K K ir ir

(30)

For nnmerieal evaluation o f d from (30), one must know the difference between / and / y, which can only be determined by the extremely difficult procedure involving radiative corrections. Thus assuming,

... (37) / +m +

K K K

we find after a little simplification*,

V / - - I f - = - - 0 - 8 9 x 1 0 - (38)

The small value o f d reflects that the real woild is approximately *Sff/(2) invariant The above solution for d (and d') abide by the c.onstraints on the isospm brealting parameteis (tfu 1973)

Next, let us examine the effe(‘t of isospin breaking on the pious It is easy to find that

(39)

The second term in the r.h.s. of (39) is positive definite, but owing i o the factor d'^ its magnitude is only 0-1% o f the first term Like //-ncson, neutral pion has no weak decay and we have no experimental information for /^ro Tint froju (39), we find,

... (40)

* We have tnkon f ^

22

P. Sinha and V. P. Gautam

At this point wc liko to mi^ntion that if our previous SU{2) approximation (37), is violated even by H)%, the rifsult (40) is altered only by 0-01% However, wo have the following sum rule from (38) and (39),

... (41) Mq (41) is not a paradoxial relation like Dashen sum rule (Haahen 1969) Tt is now s(‘(ni that the SU{2) lireaking lor the kaons is of first order in symmetry lin^alcmg intiM'aelion, whiles that for pious is of soeond order, but there is no eon- tradietion Avith Ibe geneially accepted phciumienon of octet enhancement. This moans that isospin breaking interaction is not purely electromagnetic in origin and there is some octet enhancement mechanism working, as was thought by Dashen (1972) as a resolut ion to llie paradoxial sum rule.

0

. DtSOUSSIONS and CoNCDUaTONS

Tn the pn^seut. work, we liave explored the two sourees o f mformation logard- iiig CSB, (i) pseudosealar nu'son masses and (ii) mesou-mndeon rr-torms and baiyoii masses, to iiivcwtigal.e th(‘, nature of this hi'okon symmetTy. The mixed traiislorinat ion jiropeitic^s of the t^U{'^)current wmglet, ?/p, and the *^V{^)cvrrent Octet, ?/„, undc'f SU{^)B(r„ng examined and some new insights into the (3, 3) 0 (3, 3) model of (tSB are achieved Since, Gell-Mann constant c, gives a measuri^ of the bic'aking of {'^)evrrent alone, a simple foim o f transformation from eiuTont. to eoiistitiient, quark basis, like that introduced by Fuchs (1973), was necessary to understand the relevant qudstion of tJie difference hetweoii S(J{>^)strong and U{‘^)cnrreni Even though, tlie loimalism of Fuchs (1973) has some liiiiitations, an evaluation of the parameter bja, is very much signilieant, because^ the valiums ot c and hja together provide the comi^lete mformation regard­

ing iS77(3) breaking. Tlie distinction betAVCCii the two weight parameters ‘c’

and ‘.s’, expressing the breakings of ^^b^ii)cvtrent and *S’D(3)s(ronp respectively is now understood It is also ch'iir that the difference between SU{^)i,tT0ng and *ST7(3)c«ncnir net small. In particular, tlie behaviour o f vacuum, under these two symmetry groups, brings out a significant distinction that, it is appro- ximatidy invariant under iS77(3)g(r(,„p but wof under /S’L7(3)current T^his distinction suggestis that Wignei-Ecldiart theorem can only bo applied for SU{'^)strony not lor SZJ{^)current‘

In order t.o understand the physical significance of GMOR solution we make the following plioiiomenologieal analyvsis, on which the present solution o f c, however, does not depend. Let. us consider the niesonio matrix elements of the CSB Hamiltonian, IF, expressed in the constituent quaik basis as follows,

i r - Wf,-\-swQ.

This form of H' ean bo phenomenologically written from the mixed l.ransforma- tion properties of and under SU{3)strong‘ But, in order to obtain a simple

23

relation, connecting c with s (e.g eq. 16a), w'^o need rcwsort to the coriHidcrations of section 3, The interesting feature of relation (15a), is that for exact S U {2 )^ S U {2 ) symmetry, ,v — r —'\/2. indep^ndmi of the parameters a and 6. Kow, from the SU{S)Btrong transformation properties of the wi’a, we have for the psoudoscalar mosonic matrix oloments

< P { I Wq I P j> — and < P t | | P j> — Thus, we can write.

< n I IP 17T>

< K \ I P \ K >

suit;-

(42)

(43a)

(43b)

(43c)

and Goll-Maun-Okubo mass formula follows immediately. As the pion mass- vanishes tor exact chiral *ST/(2) (g) iST/(2) symmetry (.s- - c — — ^ 2 ) we can have from (43a),

'o“ = ’ - so that it IlS easy to have liom (43a) and (‘13b),

/.> _ 1 or

Obviousl^^ the same pi'oeodure could alsf) lead to GMOR solution for o along with Gell-Mann-Okubo mass formula, if the Ui^ were assumed to transform under SU{'^)strono indices suggest. Therefore, the reproduction of Gell- Mann-Okubo mass formula in GMOR scheme, is merely a consequence of the assumed SU{^)strono behaviour of « i’s, and does not serve as a justification for GMOR solution for c Tn fact, now it is clear that, in GMOR scheme, c is identi­

fied with The other consequencci of this identification is tliat vacuum becomes approximately invariant under SUC^)current Ji'^d as a result, GMOR model fails to avido by the constraint o f Prasad (1974),

■3y(l-2/)] > 0. ... (45)- X = c /V2, y ^ c7V 2.

24 P. Sinhra and V. P. Gautam

Obviously, loi' r/ -z c, l.lis. of tlio abovo inequality relation becomes a [>(3rloot squaie, wliieh supp»)rlK our lirst postulate. By vjrluo of tbe first postulate, tlui eliii al S3uiijnetiy bi’eakin^ parameiej’s c and c' of the present scheme, always he within llie allowed domains fm them (Olcubo & Mathur 1U70, Prasad 1974).

The indep(3ndont. pj(3flietions for c and s, separately from the baryonic and tJio in(3some ('SB mail ix elements are nov^^ loimd to have very close agreement.

This iidlects tlu*, intcTiial eonsisteney of the phenomenology developed in the prescuit. analysis, i\ow, it is clear that GMOJl solution should be interpreted as tlixi ^^U{‘,i)strong breaking paiametei .s, rutlier than the ^U{^)cnrrent breaking pararnetiM’ c. flence, with this lunv outlook, OMUR solution supports the present scheme lathm than eontradiotmg. it Thus, by considering the distinction botw(}(3ii const,itiuiiil and current quarks, we have been able to overcx)iu,t the difiieulti(3S laced by GMOR model and also to recovf/r the earlier good results like eq (20) or Gcdl-Mann-Okubo mass formula

An independent, detiormmation of r based on /S't/(3) 0 /lSif/(3) gencralisa tioii ot (joMborgei-Treiman relation, c. = — ( Jones kSeadroii 1975), strongly favours the jiresent estimation (Considering |*ST7(3) 0 >S'f/(3) |t algebra, Leiitwyler (1974) lias mad<i a jirediction loi the ratio niblw, which agrees wit,h GMOR solution ilow'ever, m order to deteimine tins ratio, Leiitwyloj (1974) has estimated the individual quark masses, which depend explicitely on the Jormal vi^ctor gluon model used AVJiile, the procedure o f Hazdjian tV: Stern (1975) also involves alistractions from quark-gluon theory, but not more than the I <Sf(7(3)0 aS7/(3)Jl translormation properties o f the dyiianneal variablcy and their result is exactly relation (10a) of ours. Gunion et al (1970) have deter­

mined th(i strange and non-sti ange quark masses from meson-nucleon (r-terms, baryim mass differences and pioton Compton amplitude fixed pole value, to show that 5. which is not inconsistent with our obsorv'^ation. Individual (juai'k masses are also dotermiiU‘d by Donoghue el al (1975) in a phommienologieal analysis ol ji fixed sphere bag model, which predicts the strange to non-strange quark mass ratio nisi in 7, in exiicdlent agreement with the present estimation However, since the absolute values o f the quark masses are of no relevance in the present invi^stigation, which concerns only Avith the ratio Wsjui,, we have nothing to say n^gardmg the magnit udes of the quark masses for which widely ilifTermg lesults exist in the literatuie (Leutwylcr 1974, Hakim et al 1974, Gava it al 1975, Sazdjiari 1975, Hoiioghue cl al 1975, Gunion et al 1976),

Aiiothm- interesting and releA’-ant, point to be considered, is a comparison botAVeen the 1,avo approximate symmetries ol the world— SU{^)stronv /S(7(2) 0 1^0{"2)current Tlu'value of c can no longer tluow some light on this matter, because the departiu'o ol c from —-\/2, being a measure o f the breaking

ol' SV{2) ® SU{2)cuT7eni^ tlu* maguiiii(V of c is only a measure of the bieak mg SU{3)currcni -But, <s-is a direct jiieasuio ol the bivalung oi ajid also it IS observed that for exact aS'/7(2) ® > S f /(2 ) c ,,.s r -- — V 2, so thai like c, the deviation of s fioiii — -\/2, is also a measure of the hieakuig ol (hiial S(/{2) ^ SU{2). Thus, Irom the closeness of -v to - v"'2, n may be com-lmled that chiral SU{2) SU{2) is better eiuiserved than SU{^)stujng

Regarding other predielions from the present investigalhui, llu* sjilitliugs / „ —//f a n d /7T+—/tt" ai-e of the ordei oi tj^ucal /STA(ll) and SV{2) buakiiigs re s­

pectively and theielore, do not ajipeai to b(‘ nieon si stent. Tlu' presiM'l (l('itr- muiatioii of M^y 900 JVIeV) is also very much significant, berausi^ in tlu' limil

ifU ” > Oj f'hi.s is tlui only mass scale ami must b(i the origin ol the symnu'- tric baryon masses, the universal slope ol Rogge trajectojk^s etc. Relation (14) taken in conjunction with (15), (10) and (19), allows ns to derive- Ihe lolloAvmg hybrid moson-baiyon sum rul(\ involving il/„ hut not. the meson-baryon f7-teinis.

in p“

niS'^ irip^1 1-1 ii:!? 1 (4(i)

Mb'- and an^ the *S77(li) averages ol squaied baiyon and pseudosc'alar meson masses It may be noted tha.t tli(‘ above me„son-baiyon sum rule is indepe ndent t»f the RCA(J approximations, inv^olved in tin* punious ndalion (27) AA'lien Mji IS ri'placed by Tlfjv, (40) takes the following spi'draeular tbini.

Mb^ (40a)

The above i elation e.onipnres t lu' chiral and {‘^)Kiro„(i symmetry bHuikine effects on pions and on nucleons I'his sum rnli' may lu‘ staled in a dilhKid language as,

Ac/trai ^chiral 47)

wdiieh rellects the .stiiking symmetry bidween the non-stiangi* paitieli's ol tin- two different aS77(9)6^ronff make an ansatz that, ridation (47) may also be extended for the lundameiitul A’f^(9),snf^?ii; triplet i (- tor tire constituent quarks p. n and A, we have

whore ,so that,

4

Iiff- — I and -- {ni-„^ I 2inic]

m

(49)

26 P. Sinha and V. P. Gautam

Jf the ofteetivo Hciuarod masH mati ix Fuchs (1976b), is used to define the masseB of tlie coribtitucjit quaiks, rolatjoii (49) jkagain exactly reproduced. The value t)i as obtained independently from the meson-nucleon cr-term sin section 3, satisfies the hybrid mosun-barytm sum rules (46) and (46a) consistently. The closeness oi t(> tluJ nucleon mass imidies that among the baryons, the nueloons are hiast. ciffccted by (;hiral symmetry breaking, V'hich is again an expression of good chii al *VC/(2) (g >S'(7(2) symnuitry. As the world is mostly built up with noil-strange jiarticies. it apjiears that in th(5 real woild, there exists an approxi- matii ehiial symmetry But, in an exactly chiral invariant world, the psoudo- sealai mesons aie massless, so that short lange Yukawa lorces are absent. There- lore, l(M‘ the real uorJd to exist, chiral symmetry must be biokcii.

A CKNO WLEDGMENT

W(‘ are thankful to Norman Fuchs for many helpful communications Re f e u e n o e s

Aclomollo M & Gatto H, 1!)(}4 PIn/s. JteA> Lcit 13, 204.

Uuoc’nlla 1*^, Kleimu b H ., 8avoy G. A , Celej^hiin & ytn'acn K 11)70 Nuovo Chmenio 69A, GlU.

(Iiutui K . V & Cmt(3i A A. 1973 Niicl Phy^. B58, 378

CJxao Y A , (liilJtoHky K E , Knlly R L & Alcock J W nHS Phyt, 57B. 150 Chimg'J’. I’ , 1970 iVi.i/.v D13, 210].

JDiishoji K 1909 Pfrys. He^v,183, 1245, --- ]071 P hys Rev. D3, 187!)

--- — - 1972 Prov In t, School of P hys. JHnrico Pcrm i Coorsv 54 204 (Dovolopmout,s*iii High Enoigy PhysioK ed, K Gatto. Aciidomic Vrosa), Do A-Urtvo V , Euhmi S , Fiiilan G. & Roasoti C. 1973 C u rm its in Hadron P hysics. N oith

Holland Publishing Co, Donoghuo .) V" , Gohiwioh E & Holatom B K 1975 Phijs Reo D12, 2875

Foldy L L. Wouthuyaon 8 A, 1950 P hys. Rev. 78, 29.

Frit/soh H , Goll-Maim M & Luutwylor II. 1974 Light Cone Algebra. Ih xed M a ss S u m Rules and P C A C . Caltech preprint.

PiicliH N. J973 P hys. Rev. 11.8. 4079 --- 1970a P h ys. Rev.D l4, 1709 ---197Gb P h ys. Rev.D14, 1912,

Gatito R, 1970 S'j>ringer Tracts %n M od P h ys 53, 40,

Gu,va E,, Ijogovirii F. & Paver N. 1975 Lett. Nuovo Cimento14, 41, GolbMuun M 1902 P h ys. Rev 125, 1067,

---P h ys. 1, 03

--- 1909 Caltech pioprmt. 08-244, --- — 1972 Acta P h ys. A sir. S uppl 9, 733.

Gell-iVJmin M , Oakes R J &. Reriiioi B. 1908 P hys Rev 175, 2195, Gillman F J., Kiigler M. &> Me.sliko\ S. 1974 P h ys. Reo D9, 715 Glashow 8 & VVembeig S. 1908 P hys Rev Lett.20, 224.

Gimion .1 F , McMamoo P 8. & 8cmJroii M D 1976a P h ys Lett 63B, 81.

--- 1970b 8LAC-PCB-1847, (now publ hod in N uel. P hysB123, 445 (1977))

Hnkim H .1,, Logovim F. & Pavpr N 1974 LcU. Nnovo. Cimrnto 9 ,17!),

Hoy A J. (,} 197() Quarks and Symniefru's PTcprimt—iSouthampton Univovsity (hoc also tho bibliogmphy oompilod by J. M. N. Hoy at tho oml of this arfiolo), Hoy A J G , Rosner ,1, L, & Weycrs J. 1973 Nuvl Phys B61, 205

Hito G. E, & Jacob R . J. 1974 Phys. Lett. 53B, 200, Hu B. 1973 Phys. Lev. D7, 24B3,

Jonoa H. F. & Rcadmn M. D. 1975 Phys. Rev. D ll. 174.

Laii|£;boin W. 1975 N'ucl. Phys B94, 519.

Loiitwyloi' H. 1973 Nucl. Phys. B76, 413

MoNaineo V. C. & Soadron M. D 1975 Ntiovo Cimerito 30A, 287 Moloali H. J. 1974 Phys Rev D9, 1095

Moir D. C , Jacob R. J & Hito G E 1976 Phys. B103, 477 NiolaoTi [J. A. Oacloa G, C. 1974 JVw(7 Phys B72, 310.

Olmbo S & Mailmr V S 1970 Phys. Rev. D l. 2046 3468 Paitiolo Data Group 1976 Rev Mod. Phys. 42, No 2, Pi. Tl.

Piasart S. O 1974 Phys. Rev. D9, 500.

iSazdpan H. 1975 A nciv eMimate of the. Quark masses ORSAY preprint, (IPNO/TH 75-28) Sazdjian H. A Stern .1 1975 Nuel Phys B94, 103.

Scadron M .D .rfc Jonoa H. F. 1974 Phys, i?ci>,IDIO, 967.

Smha P A Gautam V P. 1974 hcM, Nuovo, Gimento 11, 83.

--- --- 1976 Jnd J. Plmjs. 50, 981.

Sutherland D. G. 1966 Phys. Lett 23, 384.

von Hippol F A Kim J K 1970 Phys Rev. D l, 151,

Note added in proof :

I f OHO nouMiders the matrix elements o f IP expressed m constituent quark basis between quark states and takes into aecoimt tht‘ vanishing oi non-strange quark (bare) masses in SU{2) 0 SU(2) symmetry limit, it can bo sliown that

./2 2 m fT (50)

from vyhi(‘h relation (48) direclly lollows

In our discussion on isospin breaking in section 5, we ha^'^e considered the effect o f the dominating tadpole term [uf) only, while the non-tadpole contribu­

tion duo to «J J-typi‘ e.m. interaction has been neglected. Along with our postu­

lates, r; — o' and d — d'. an account of JJ-type e in, contribution to isospm breaking, leads to equations like.

[^m J - ) j j = (w

K h /t a » J jj{ (51)

(52) Tf wo take,

(Am ^2_ Am o“)jJ ^ by Dashen’s theorem (Dashen 1969), we get

d::^ - M O x l O-2 (53)

for f .

28 P. Sinha and V. P. Gautam

|j] (>nl(‘) !■) flan.'^loi'in Hu*- (Sll) of //', to constituent quark l>’r;is II Iciiii tiiu\ Im' |)li(inr>nirMoJoj4')caJlv iiiclu(l(‘(l in the (‘xpr(*ssion foi V '(1 ( l l j Tli(‘ vv(M'j,lii ])fiimnctei associided witli tins tenn, relative to the i.enii m U, will ul nrnisr*, ho v(mv siniill f?) In this ease H' can be VTitten m the eoiistitiu'iil. (jiiark ()ukis, as

/ /' - n\i 1 ... (54)

Thus, all (he jiit(‘i-iiiiiltijik 1, mlia-multiplol and hybrid sum rules of Coleman (V (hasliow (IDOr)) imiiaiji in tnet In this eonueetion it may be lemarked that III the SIJ{'2) intis'^ s|)lil lin^^s jqipearnu/; m fhe (hleman-Clashow sum rules, the l.arl|ml(‘ roiitiilml jons aloiu' should ajijif'iii, ratlim (han thi^ exiierimeutal mass Hjililt.iMii;^ - 'I'lu si^ iad])ole eonti ilmtions may ho obtained by subtracting; the non t,ad|)olo »hJ-tyj)o eoiitnhulion to NP(2) sjilittings from the expei'iimmtal oi'os. Ho\eev(‘i, evim (iKm^li, sotiu' estimations Ibi such JJ-type e.m mass s|)jittiii!;s, 1‘xjsl in the lileratuK' (F liuecclla et al 1969, R H. Soeolow 1965), they (‘an not be taken too si»i iously due to the approximations involved.

Idle latio /-/.y ean be obtained by the use of Wigner-Eckart theorem, as loiloAVS

1 ^

'' -

'Pho filioA'o solution ean also lu' reeovmcMl, if oik‘ sohn‘s for fhe ratio dje with the CMOK. nwti letioiis thaf.

— 0 and i e..

f^ M O R (50)

('G M O ll

m consistent with oni interxiretai ion of cr,-mo/;

It may be iiotiul that the ratio

tfs,

appearing in (55), is the same as

:ij'

> ol (^okmnin c\r CJashow (1965) or

c/jc

of McNainee

ft

(1975) ol y ol (Juiiion

<‘i al

(1976b), How(‘V(‘r, we feel that, in the work o f Giinion

el al

(1976b), according to the doliiiition of rvj and through their eq (5.1), the ratio ':i/^n ( A') should correspond t,o the ratio dje while, if the subscript ‘GMOR’

IS \vritl.(m correet.ly,

X

gmof should em-respond to the ratio

Ijs

Similarly, in (ujs (S) or (5J) oi McNamee

et al

(1975), it has been incorrectly taken that fhe iclat iA7‘ weight pa.iaimder associated AVith the t3U{ii)sirong breaking part H (Ag), is c, instead ol .v or

We arc‘ thankful to Mike Scadron foi a stimulating letter

(55)

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Cot man S &; ({lashow S 11)65 iVii/.s Ifcr134B, 671 Soi'olow d H J965, PAi/,s'. /frr 137B 1211.