Volume 40B, number 2 PHYSICS L E T T E R S 26 June 1972

R E A L I Z A T I O N O F S C A L E AND C H I R A L S U ( 3 ) × S U ( 3 ) I N V A R I A N C E W I T H G O L D S T O N E B O S O N S AND B R O K E N S U ( 3 ) C O U P L I N G C O N S T A N T S

S. ELIEZER Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, UK

and

R. DUTT Department of Physics, Westfield College, University of London, England, UK

Received 14 May 1972

Present exoerimenta[ evidence for dfiaton (¢ meson) is consistent with unsubtracted dispersion relation of (P]0 , ~, [P> (P = ~" ,kJT) dominated by two pores and SU(3) viotation in £PP couplings.

A scale invar ian t theory is charac te r ized by the conserva t ion of the dilat ion opera tor D, which may be defined in t e r m s of the ene rgy -momen tum tensor Ogv [1,2]

D = fd3xx ~O~to(x) . (1)

When the symmet ry is not rea l ized in na ture (i.e. dD/dt¢ 0), one has from eq. (1)

~z = 0u~ (2)

where -9 v = xg 0, , i s the di lat ion cur ren t . Gell- Mann has con jec tures [1] that the 0god = 0 l imi t is cor re la ted with the exis tence of a m a s s - l e s s sca la r par t i c le , the S o meson. This is in analogy with Goldstone-Nambu rea l iza t ion of SU(3) × SU(3), where the en t i re pseudosca la r octet has zero m a s s [3].

When scale or di la t ion symmet ry is broken in na ture , it i s hypothesized that the s ca l a r i so- sca l a r mesons dominate the mat r ix e lements of the t race of the ene rgy -momen tum tensor O~v. This is p rec i se ly what the hypothesis of pa r t i a l conserva t ion of di la ta t ion cu r r en t means . In a recen t work, C a r r u t h e r s [5] a s sumes unsub- t racted d i spers ion re la t ion for the hadronic form factor due to 0~ together with the hypothesis of pa r t i a l conservat ion of di la tat ion cu r r en t and ob- ta ins a G o l d b e r g e r - T r e i m a n type re la t ion , in- volving m a s s e s and couplings of pseudosca la r mesons and baryons in the form

/So GSoPP + fs8 GssPP -- (3)

/S O GSoBB +/s8GSsBB = M B , (4)

where the coupling/Si is defined as

(0 }OgvlSi(P)> (2wp) I/2 L p _ = 3fsi(P~ v g~p2) (5) so that

<0[O~{Si(P)>(2wp)l/2 = _fsim2 i ( i = 0 , 8 ) .

However, C a r r u t h e r s calculat ion with unsub- t racted form of F(q 2) --- <~I0~ ]~> does not sat isfy the slope cons t ra in t as imposed by Ward Identity ana lyses [6].

An a l te rna t ive form of eq. (3) has been der ived by severa l authors from different approaches [6-8] to sat isfy the requi red slope condition. Using Ward- l ike ident i t ies Kle iner t and Weisz [6] necess ia te a subt rac t ion in F(q 2) a s suming that 0~ i s dominated only be the s c a l a r - i s o s c a l a r smglet ~- (So). Thus they obtain

/So GSoPP = m2so + (~- 2)m2 (7) where d u is the scale dimension of the scalar densities u which break chiral as well as scale symmetries in the energy density

Ooo = ~oo + ~ + E (u o +cu 8) . (8)

0oo is ch i ra l and scale invar i an t and 5 is ch i ra l i nva r i an t but b reaks scale and con.formal i nva r i - ance. Eq. (7) p red ic t s an abnormal ly la rge width F(~ ~ ~ ) -~ 1200 MeV provided one uses f A = 102 MeV as obtained by C a r r u t h e r s [5]. A1Ss ° it gives a la rge value to the form factor FuS ° de- fined by

(So(k)IAg(0)]~(P)) (4w k cop) 1/2 =

i[(k +p)g Fuse(q2) + (k - P)~z Gyso(q2)] " (9)

and makes it incompat ible with the cons t ra in t

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Volume 40B, number 2

FuS <~ ½"/2 imposed by the A d l e r - W e i s b e r g e r sum°ru le for nn sca t t e r ing [9]•

As an a l te rna t ive p ic ture of Kle iner t and Weisz, we propose here that eq. (7) can be der ived by as - suming unsubtracted dispersion relation for F(q 2) but with two poles dominance (S o and S 8) of 0~u I

In this mechan i sm eqs. (3) and (7) can be sa t i s - fied s imul taneous ly , s ince des t ruc t ive i n t e r f e r - ence takes place between S o and S8~pole con t r i - but ions to the mat r ix e lement F(q ~" =0), which is of o rder E (see eq. (3)). As far as the slope dF/da2I~2=0____, is concerned, there is no d e s t r u c -

'4 tive in t e r f e rence and only S o gives the dominant contr ibut ion, which is of o rder unity, sa t isfying slope cons t ra in t [6].

We fur ther invoke SU(3) violat ion in SoPP (P = n ,k , ~7) ver tex for cons is tency of our equa- t ions. This assumpt ion may be motivated by the following a rgumen t s : Using the hypotheses of pa r t i a l conserva t ion of ax ia l -vec to r cu r r en t to the m a t r i x e l e m e n t s ( P 1 A . Is> w h e r e t' = , , k or 77 and S is the appropr ia te sca l a r meson, one ob- ta ins for the s c a l a r - p s e u d o s c a l a r - p s e u d o s c a l a r couplings

/ ~ 2 m 2 GSp p ocv,~ S- pj. (10)

Eq. (10) indicates that Gc v,v, vanishes in the oO.t- r

l imi t of exact ch i ra l and scale invar iance , where- as it acqu i res f ini te value when the s y m m e t r i e s a re broken. In the ch i ra l model of Gel l -Mann- Oakes -Renne r [3,4] it i s rea l ized that SU(3) and ch i ra l SU(3) × SU(3) s y m m e t r y breaking i n t e r ac - t ions may have comparable s t rengths , and hence working in this model it i s not unreasonab le to expect a s izable cont r ibut ion to G S p p due to SU(3) breaking alone. The same a r ~ m e n t , how- ever , does not apply to the coupling GSspp , which r e m a i n s f ini te in the l imi t of scaIe and ch i ra l s y m m e t r i e s . So one may poss ib ly ignore SU(3) violat ion in it. In genera l , SU(3) violat ion should not be neglected in coupling constants if al l three pa r t i c l e s a re Goldstone-Nambu bosons. Thus we wri te

GS8Pi P j = g8 dsi j ( l l a )

G S o P / ~ = g o 6 i j + h d 8 i j ( i , j = 1 ,2 , . . . , : l)lb )

where g a re the s y m m e t r i c p ieces and h is the octet broken par t of the coupling constants .

Let us now cons ider the analogue of eq. (7) genera l ized for the pseudosca la r octet

rn 2 + ( d u _ 2 ) m 2 (P = n,k,7?) fSo GSoPP = So (12)

In der iv ing eq. (12) we have ut i l ized the Ward ident i ty

PHYSICS L E T T E R S 26 June 1972.

-iq iz T~(q,p) = T(q,p) +i fd4 xexp{i(p-q)x} 6(x o) x

×{01 [Ao(0) , 0 v (x)][ P(p)) (13)

where

T~t(q,p) -= i fd 4 xexp(iqx) (0 [T{O~(O)Att(x)}[ P(p))

T(q,p) =- i f d4x exp(iqx)(0[ T{O~(O)~ZAg(x)}{ P(p)).

(14) Now invoking the hypotheses of par t ic le conse r - vation of delatat ion cu r r en t and par t ic le conse r - vation of axial cu r ren t , which mean that the mat r ix e lements of O~ _ and ~/~ A~t are dominated by sca la r and pseudosca la r mesons respec t ive ly , and using the commuta tor [7]

[QA,6L~] = i ( 4 - d u)OgAg (15)

we obtain in the l imi t qg --* 0 : 2 2

/SomSo fS 8 m S 8 G - -

So (16)

Following Crewther [8] we may der ive

/SoFpso = /p +O(6,e). (171

Using the re la t ions

fp GSpp

par t ic le conserva t ion of axial c u r r e n t

= (m2 s - m )%s (18) together with eqs. (3), (16) and (17), we obtain eq. (12) neglect ing second and higher o rder t e r m s in E and 6. Equations (3) and (12) give the following solutions: 7q~ 2

SO : ~(2m2k+m 2n) (4 -d u) (19a)

(l b)

fSogO = ~(2m 2 + m 2 ) . (19c)

By demanding maximal smoothness , our eqs. (12), (17) and (18) give

d u = 1 , (20)

cons is ten t with the recen t work of Jackiw [10]. This solution supports the necess i ty of SU(3) breaking in G S p n because if there would be no

• O ~ - spl i t t ing i .e. h = 0 in eq. (12), we would have m• = m k = m f o r d u ¢ 2. Using d u = 1 i n eqs. (19a) and (19c) we have*

* We have neglected the effect of particle mixing both for scalar and pseudosca[ar mesons and so the field So(X ) correspond to the physical particle e(700). The existence of a scalar-isoscaiar with mass 1200 MeV is compatible with this assumption [11].

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Volume 40B, number2 P H Y S I C S L E T T E R S 26 June 1972

mSo-= m~ = 714 MeV (21a)

h i e o = 0.77 , (215)

showing r e m a r k a b l e a g r e e m e n t of the p r e d i c t e d m a s s of s c a l a r c wi th i t s e x p e r i m e n t a l va lue 700 MeV [12]. The r e s u l t a l so d e m o n s t r a t e s that SU(3) sp l i t t i ng in G S _ p p coup l ing i s qu i te l a r g e u c o m p a r e d to i t s s y m m e t r i c va lue . A s i m i l a r r e - su l t i s ob ta ined in a r e c e n t c a l c u l a t i o n ba sed on the GMOR m o d e l [13].

Now we p r o c e e d to d e t e r m i n e f s ° f r o m the low e n e r g y sum r u l e [8]

2 2 = fso so +

Using the s t anda rd d e c o m p o s i t i o n (eq. (8)) of the e n e r g y dens i t y and a s s u m i n g 6 to be a c - n u m b e r [14] ( i .e . d 5 = 0) we have the " v i r i a l - t h e o r e m " [1]

0~ = E ( 4 - d u ) ( u o+cu8) + 45 . (23)

E q u a t i o n s (22) and (23) g ive

is2ornS2 ° + fs8r n 2 2S8 = e d u ( 4 - d u ) ( ~ o + C ~ 8 ) , (24) w h e r e we have used the de f in i t ion ~i -= <01uT10>, i = 0 , 8 and the c o m m u t a t o r

[D, Ui] = - i d u U i • (25)

To l o w e s t o r d e r in SU(3), we obta in f r o m eq.(24)

f 2 m 2 = d u ( 4 - d u ) e ~o (26) So So

In the GMOR m o d e l , the v a c u u m e x p e c t a t i o n v a l u e ~o and ~8 m a y be r e l a t e d to f k and f n v i a low e n e r g y sum r u l e s [15,16]

: ( 2 7 a )

: ° - (275)

and c i s g iven by [17]

Ilk 2 m2"~ ,//k 2 i_ 2 k - .'/t mk + : . s " 1281 ~S.

We e v a l u a t e SS,~ f r o m eqs . (26) to (28) t r e a t i n g f k / f y as an inp~it p a r a m e t e r and u s i n g f ~ = 94 MeV. F r o m eqs . (18) and (19c) we d e t e r m i n e F n S and go and p r e d i c t the width of E ~ 2n decay g ive~ by **

2 G~n7 r 3P n

(29) F(E ~ ) - 4n 4m2

E

• * We have defined GSoPiP] (460pi 60~ )1/2 : <S O I J ; i iO]> where j+ = (E]+m2)qSiis the meson current. If we define the Lagrangian 2 S o ~ = gSo~rTr So q~2 we have

the connection GSo~rTr = 2gSoTr~r .

w h e r e

G ~ = GSon~ = go(1 1 h + (30)

The r e s u l t s a r e

S~ c / S o FnSo(0) (MeV) (MeV)

1.1 -1 .27 124 0.77 600 1.2 -1 .28 135 0.71 505 1.28 -1 .29 142 0.67 460

F ina l l y we conc lude that the p r e s e n t e x p e r i - m e n t a l da t a fo r the d / l a t ch ( m a s s , width , e tc . ) a r e c o n s i s t e n t with: a) u n s u b t r a c t e d d i s p e r s i o n r e l a t i o n of ( P i O ~ i P ) a s s u m i n g i t s d o m i n a n c e by two p o l e s , b) s i gn i f i c an t SU(3) v i o l a t i o n in S o P P coup l ings and no a p p a r e n t S o - S 8 mix ing .

The a u t h o r s a r e g r a t e f u l to P r o f e s s o r B. R e n n e t fo r f ru i t fu l c o m m e n t s and s u g g e s t i o n s . One of us (S. E. ) would l ike to thank Dr . S. P . de Alwis fo r s e v e r a l use fu l d i s c u s s i o n s . R . D . w i s h e s to acknowledge f i n a n c i a l suppo r t f r o m the S. R. C. and S . E . i s g r a t e f u l to the Roya l Soc ie ty P o s t d o c t o r a l F e l l o w s h i p exchange p r o g r a m with the I s r a e l A c a d e m y of Sc i ence .

Re ferences

[1] M. Gel/-Mann, Hawaii Summer School Lecture notes (1969).

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Rev. 175 (1969) 2195. [4] S. L. Giashow and S. Weinberg, Phys. Rev. Let ters

20 (1968) 224. [5] P. Carruthers , Phys. Rev. D3 (1971) 959. [6] H. Kieinert and P.H. Weisz, Nucl. Phys. B27(1971)

23 and Nuovo Cim. 3A(1971)479. [7] S.P. de Alwis, Nucl. Phys. B28(1971) 594. [8] R.J . Crewther, Phys. Let ters 33B(1970) 305 and

Phys. Rev. D3 (1971) 3152. [9] S. L. Adler, Phys. Rev. 140B (1965) 736;

F. J. Gilman and H. Harari , Phys. Rev. 165 (1968) 1803.

[10] R. Jackiw, Phys. Rev. D3(1971) 1347. [11] B. T. Carroll et al., Phys. Rev. Let ters 28 (1972)

318. [12] Par t ic le Data Group, Phys. Let ters 33B (1970) 1. [13] R. Dutt, S. Ei iezer and P. Nande, Phys. Rev. D4

(1971) 3759. [14] J . Ellis , Phys. Let ters 33B (1970)591;

J . Ellis , P .H. Weisz and B. Zumino, Phys. Let ters 34B (1971) 91.

[15] B. Renner and L. P. Staunton, DESY preprint 71/62, [16] R. Dutt and S. El iezer , Phys. Rev. D4(1971)180.

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