P H Y S I C A L R E V I E W D V O L U M E 1 4 , N U M B E R 1 I I D E C E M B E R 1 9 7 6

K*0(890) decay density matrix elements and partial conservation of strangeness-changing vector current

Kashyap V. Vasavada Department of Physics, Indiana -Purdue University, Indianapolis, Indiana 46205

(Received 23 February 1976)

We show that the zeros in the density matrix elements pj, _ , and Rep;, for K p 4 K*'n ( K ' n - K*'p) are correlated and imply fundamental relationships among the Ball amplitudes following from partial conservation of strangeness-changing vector currents and the smoothness hypothesis.

In a recent paper1 we considered pO-meson de- cay density mat r ix elements in the p rocess n-b - p0,2 and found that two of these , P:,, (Gottfried- Jackson f r a m e ) and Re pf, (s-channel helicity f r a m e ) , had remarkable energy-independent fea- t u r e s . pi,_, is consistent with being z e r o f r o m

1 t I = 0 to at least up to 8711; f o r incident pion lab momenta of 2.7 G ~ V / C to 17 G e ~ / c . Reps, goes through z e r o a t t= - 171; over the s a m e energy range . Although the z e r o of Reps, can be obtained in a gauge-invariant (electr ic) Born inode12t3 o r absorption models (especially the Williams mod- el4) , we found that these z e r o s imply specific, to some extent, model-independent relat ions among the invariant Bal l amplitudes fo r the p rocess . p:,, -=O together with the sllloothness relat ions of Cho and S a k ~ r a i ' ~ ~ and Achasov and Shestakov5 led t o the z e r o of Rep:, a t t= - 112," in a natural way without using any specific exchange model.

Recently high-statistics data f o r the correspond- ing s t range-part ic le react ions K-p -R*O(890)tz and K'n -K*'(890)p became a ~ a i l a b l e . ' ~ We do not exhibit the data h e r e s ince they have been already

published o r will be published in the near future. Also pj,., and Rep;,, with which we will be mainly concerned in this paper , have almost identical fea tures to the p0 data presented in I . One again has pi, = 0 f o r a l a rge range of energy and extend- ed range of momentum t r a n s f e r , and Reps, p a s s e s through z e r o a t t = - I J ? , ~ . If we pursue a n ana1ysj.s s ini i lar to I , interestingly enough, the z e r o is found near 1 = - 111 ,' (K- meson n ~ a s s ) ~ and not t = - 111; s ince in this analysis only the external par t ic le m a s s e s e n t e r . The shift in the z e r o f r o m - J ~ I , ~ to - 111; is explained h e r e in a natural way by invoking par t i a l conservation of s t range- ness-changing vector c u r r e n t s (PCVC) ."lo PCVC has been used extensively in I<,,-decay a n a l y s i ~ . ~ ' ~ ' ~ It is very interesting to find i t s confirmation in purely s t rong processes such a s the p resen t one.

Since essent ial ly a l l the equations of I (except the ones explicitly mentioned here ) hold f o r the present work with minor changes (111; - ~ I I ~ * , ' ; p 2 - i71K2) we will not repeat them.

The mat r ix element fo r the p rocess K'(q) + p ( P I ) -

R*O(iz) + n ( p ) is given by

+Be(< . P ) ( y a Iz)+B,(E * k) (ye 1z)+B8(< q ) ( y q k)]z/(fi,h,). (1) r --

J E * is the source cur ren t and E is the polarization Now, if the Ball amplitudes a r e independent of vector f o r fC*O and P = $ ( p + g l ) . Note that accord- k2, one obtains the following ( incorrect) relations: ing to the notation of Ref. 3 , which we a r e using, p' and p a r e the momenta of the zlzitinl and fztzal sB, - (1 - 171,~)B, = 0 , (3) nucleons, respectively. Then s = ( p + k)' = (p'+ q),, t = (p' - = (q - 1 ~ ) ~ . B l + B , + 2 B 4 = 0 . (4)

If we assume that R*' is coupled to a conserved cur ren t ( a , ~ E * = 0 ) then a s in Ref. 3 we a r e led to These a r e the so-called smoothness relat ions. the following ( incorrect) relation (in the high-s Then, following along the s a m e l ines a s i n I , l imit) : we find that p:.-, = 0 would requ i re Bl = - B,. (Ac-

k ' ~ , + sB, - ( t - 111,' - P ) B 3 + 2k2B, = 0 . tually the equgtion is somewhat different, s ince (2) although ,112 is negligible compared t o 111,2, in," - ,

T h e r e is a s imi la r relation between B,, B,, B,, is not s o when compared to iii,*? A m o r e care - and B, but these amplitudes a r e not needed in this f u l t reatment is given in Appendix A , ) This re - paper . suits in the amplitude D- (defined ill I) a s

3112 K A S I I Y A P V . V A S A V A D A - 14

In deriving Eq. ( 5 ) t e rms of the order of l/s have been neg1ected.l" Also a s in I, the contri- bution of the B, tern1 has been justifiably neglected. F i r s t of al l from exchange diagrams i t i s seen to be small . Also the coefficient in front (2111 against s for B,) is small . The coefficient of B,(f) i s also sinall but B, receives a dominant one-n-exchange contribution with a very sinall denominator. Using Eq. (3) we see that D _ passes through zero a t i s - 111,~. NOW E q. (14) of 1 gives

6 Rep;, = Re(D+T: + D-T?'). (6)

Again the D+T$ (nucleon-nonflip) t e rm can be expected to be very small a s compared to the p-2': (nucleon-flip) t e rm. Hence Rep:, will pass tltrough zero at 1= - 11rK2 in complete disagreement

I

where T K - P - ~ ~ ~ s the matrix element for the process KIP - p n , which can be defined by

(11)

Omitted t e rms in Eq. (10) iilvolve y,y, with B 5 , B, , B,, and B,. Now we assume that B ' s a r e in- dependent of k', set k2 = 0 , and obtain

Writing the B, te rm a s

we see that the (correct) smoothness coildition can be written a s

pfovided that (111,~ - III,")CY,+/III,*~)B, agrees with the right-hand side of (12).

Now consider the couplings of these currents to the K-a system. This would then correspond to taking the one-pion-exchange contribution to E,. Baryon poles do not contribute to B, appreciably. From Appendix B

Hence the cancellation of n-pole parts requires the conditioil

with experiments .Gs This difficulty can be immediately traced to the

current-conservation condition which led to Eq. (3 ) . As pointed but in Refs. 9 and 10 and subse- quent works by several authors, because of the breaking of SU, symmetry we cannot expect con- servation of the strangeness-changing vector cur- rent . However, on the positive side, a s with PCAC, it would be partially conserved (or "con- served a s exactly a s possible"). Then we have

ailsl, =g,M,"x, (7 )

where S, i s the strangeness-changing vector cur- rent to which the R*' meson i s coupled. K i s the K-meson field operator. g, i s defined by

(0 1 sl, 12') = LgKk,, (8)

Similarly g,+ can be defined by

(oIs, IR*O)=~,*E,. (9)

Then instead of (2) we have

But this i s precisely the condition obtained (Appen- dix C) from application of PCVC to the matrix element of S, between I< and n states.

Another equivalent way of looking at the result follows from the original paper on PCVC .' The illlplication is that the K*-production illatrix ele- ment develops a pole in k2 at k2 = II~:, and for zero mass K this will prevent B, from being smooth in 12. This gives

where the coupling f i s defined by the Lagrangian density f I<:a lL~ . The current-conservation condi- tion now becomes

(18)

where B, does not have a K pole. Reference 9 gives a Goldberger-Treiman-type relation (in our nota- tion)

In the limit 111; - 0 , the f terin again changes the coefficiellt of B! from - ( I - 1uK2 - k2) to - ( t - ~ 1 1 , ~ - k2) and thus the condition (14) is obtained by as-

14 £ * ° ( 8 9 0 ) D E C A Y D E N S I T Y M A T R I X E L E M E N T S A N D . . . 3113

s u m i n g independence of k2. In both m e t h o d s , we have shown th i s c ance l l a t i on

fo r the (dominant) p ion -po le p a r t s of B3 and A. In Appendix B we show tha t the s m o o t h n e s s condi t ion (14) and Bt = - B2 a r e sa t i s f i ed by the b a r y o n - (A0, S°) pole con t r ibu t ion to B2 and the p ion-po le con ­t r i b u t i o n to B3 if S a k u r a i ' s p r i n c i p l e of u n i v e r s a l ­ity of p c o u p l i n g s , p - m e s o n c u r r e n t c o n s e r v a t i o n , and SU3 s y m m e t r y with F - t y p e coupl ings for v e c -t o r - m e s o n - b a r y o n - b a r y o n v e r t e x a r e used . 1 3 T h u s in such m o d e l s the c o r r e c t condi t ion (14) can be r e a d i l y u n d e r s t o o d . Equa t ion (14) t o g e t h e r with B1 ~-B2 p r o d u c e s a z e r o in Dm at t~ - m2 which in t u r n g ives r i s e to a z e r o in Repf0 n e a r t h i s po in t .

In a b s o r p t i o n o r R e g g e - c u t m o d e l s one would obta in t h i s z e r o by modifying 7i-pole d i a g r a m s . T h i s aga in qua l i t a t ive ly we can u n d e r s t a n d t h i s effect in t e r m s of P C V C .

A s for the e x i s t e n c e of K a s a bona fide p a r t i c l e , t he e x p e r i m e n t a l s i tua t ion i s s t i l l not c l e a r . 1 4

E i t h e r it i s a r e s o n a n c e with v e r y l a r g e width o r jus t a c o m p l i c a t e d K- IT s - w a v e phenomenon in which the p h a s e shift m a y o r m a y not be r i s i n g t h r o u g h 90° in the r e g i o n of 1200 to 1400 MeV. Since o u r d i s c u s s i o n d o e s not depend on the ac tua l va lue of mK} gK, o r gKKt we can jus t r e g a r d the r i g h t - h a n d s ide of the PCVC r e l a t i o n a s s o m e ef­fec t ive way of r e p r e s e n t i n g S U 3 - s y m m e t r y b r e a k ­

ing by th i s s - w a v e p h e n o m e n o n . F ina l ly we note tha t m o r e a c c u r a t e t r e a t m e n t

g iven in Appendix A g ives the z e r o in Repf0 a t £ = - 0 . 5 3 m2. It i s v e r y i n t e r e s t i n g tha t the r e c e n t e x p e r i m e n t at SLAC6 g ives t he z e r o a t t~ ~ 0.011 GeV 2 , which i s r e m a r k a b l y c l o s e to o u r va lue - 0 . 0 1 0 GeV 2 . In p p roduc t ion the c o r r e s p o n d i n g z e r o o c c u r s a t t~- m2 ~~ 0.019 GeV2 and thus t h e r e i s a shift t oward a s m a l l e r | / | v a lue in the K* d a t a in a g r e e m e n t with o u r a n a l y s i s .

T h u s we can jus t i f iably conc lude tha t the z e r o s in pi ml and Repf0 i nd ica te con f i rma t ion of the s m o o t h n e s s condi t ion and p a r t i a l c o n s e r v a t i o n of s t r a n g e n e s s - c h a n g i n g v e c t o r c u r r e n t . To a c e r ­t a i n extent t h e r e a r e h in t s of m o d e l - i n d e p e n d e n t p r o p e r t i e s of the fundamenta l a m p l i t u d e s of such p r o c e s s e s involving v e c t o r m e s o n s .

I w i sh to thank D r . E . F i s c h b a c h and D r . F u c h s for helpful c o n v e r s a t i o n s .

N .

APPENDIX A: MORE PRECISE LOCATION OF

THE ZERO OF Rep >10

In the tex t , it w a s men t ioned that , un l ike the c a s e of p p roduc t i on w h e r e m2 i s neg l ig ib le c o m ­p a r e d to w p

2 , mK2 i s not so when c o m p a r e d to mK*2

in the p r e s e n t c a s e . Hence the equa t ions have to be s o m e w h a t modif ied. p,/„i = 0 s t i l l l eads to the equat ion [Eq. (13) of l]

(1 - c o s # ) ( - sB,) + sixvc ( - sB9) + 2/ cosx + sin%v • (t +mK*2 - mK J3, = 0 . (Al)

But the c r o s s i n g ang l e s a r e given in a b e t t e r a p ­p r o x i m a t i o n by

c o s # = -mK*2 - mK

2 +t

m»>* - w« t(mK*2 + mK

2) {mK*2-mK

2)

and (A2)

S M = -2yPlmK*

2 t{mK*2 + mK2)

K K (mK* - mK)

The coeff icient of J53 s t i l l v a n i s h e s , but exac t c a n ­ce l l a t ion of the BlyB2 t e r m s d e m a n d s that

2

£ , = - — £ 2-JL-B9 = ^0.W2 (A3)

i n s t ead of BY = -B2 a s in the p r e v i o u s c a s e . T h e n Eq . (5) g ives

1 (™x* - * " * » _ 2 a .

Us ing s m o o t h n e s s Eq . (14), we ob ta in

(A4)

2t B,, (A5)

which gives z e r o a t

2(mK*2-m/) • m„ n (mK*2 + mK

2)

(A6)

F r o m Eq. (6) it follows that the z e r o in Repf0 w i l l b e obta ined c l o s e to th i s t va lue .

APPENDIX B: BARYON-AND PION-POLE TERMS

T h e s - c h a n n e l {K~p~~K*°n) A po le and the t-channel (K~K°*~-*np) TT po le a r e given by

SKNKSK:¥NMP,)S^V 2 [ (w A - m)(y c) - ( y € ) ( y k)

+ €• & + 2c- P + €- q \u(p),

(Bl)

(B2) , ., ( 2 € - t f - C / e ) . v

5 t - m„

3114 K A S H Y A P \ r . V A S A V A D A - 14

T h e s -channe l C0 pole g ives a s i m i l a r contribution. C h a r g e l abe l s have been s u p p r e s s e d . F o r l a r g e s,

and

Mow the ~*,\~h, K*~V?? coulslings can be r ead i ly r e - la ted to the K*KT couplings by us ing SU, s y n l m e t r y with only F - t y p e coupling f o r vec to r -meson-ba r - yon-baryon ve r t ex and u n i v e r s a l couplings of the

p meson . A l so K X A , IfNZ couplings are s in l i l a r ly

r e l a t ed to nh'ih' coupling fo r any value of the F / D ratio. '" It i s found that

f o r the a p p r o p r i a t e cha rged s t a t e s and any F/W

r a t i o f o r p seudosca la r -meson-ba ryon-ba ryon v e r -

tex. F r o m (B3) and (B4) it is eas i ly s e e n that the

t h r e e t e r m s sa t i s fy the slxioothness Eq. (14) wi th

( 1 - 111,~) as the coefficient o l B,. In a s i m i l a r way, the s - c h a n n e l (A, 2:') and t -

channel (71) pole t e r m s f o r K-p .- FYI a r e g iven by

'I<. V . Vasavada and L. J . Gutay, Pllys. Rev. D 2, 2563 (1.974). This paper will be referred to as I hereafter.

2C. F. Cho and J. J. Sakurai, Phys. Rev. D 2, 517 (1970).

k6. F. Cho, Phys. Rev. D 4 , 191 (1971). "P. K. Willianis, Phys. Rev. D l , 1332 (1970). "J. N. Achasov and G. N. Sheslaliov, Yad. Fiz. Kt, 1090

(1970) [Sov. J. Nucl. Phys. _11, 607 (19'70)l. 6 ~ . W. Brandenburg ct a1 ., Phys. L,ett. -%, 405 (1975).

Some previous references a r e given here. 'A. B. Wiclilund et u l . , in Proceedings of the XVilI7z-

ternntional Conference on High Erzergy Physics, Lon- don, 2974, edited by J. R. Smith (Rutherford Labora- lory, Chilton, Didcot, Berlrshire, England, 1974), p. 1-176, and private communication. I wish to thanlc Dr . S. L. I(vamn1er for sending the data of this group prior to publication.

' D . Cords et a1 ., Phys. Rev. D 5, 1974 (1971). 9. Nambu and J. J. Salnirai! Phys. Rev. Lett. 11, 42

with a s ln i i l a r t e r m fo r CO. Note that the baryon-pole contr ibut ion to the "A"

t e r m of T ,-,_, o,, goes down by a fac to r of 1 , s as compared to the a-])ole t e r m . Hence ~t can b e con-

s l s t en t ly neglected.

IU'PENIIIX C. PCVC BETWEEN K AND n STATES

A number of a u t h o r s have used PCVC in connec-

tion wit11 K,, decay.'"'2 T h e m a t r i x e lenlent of S , between K and a s t a t e s i s given by

( r iq2) sir lR(q1)) =i [ f7( t ) iq1 +q2) j l+ f - i t ) (q l - qJu1.

(C1)

Hence

( ~ ( 4 ~ ) 1 a,s, liiiq,)) = f , ( t ) ( ~ ~ , ~ - 1 1 ~ ~ ) + t - ( t ) t

=fit) iC2)

with 1 = (q, - q2)2 . If w e u s e PCVC Eq. (7) and as- s u m e that the f o r m f a c t o r s obey unsub t rac t ed d l s -

p e r s l o n relations, saturation of f+ ( t ) with K" and f ( 1 ) -71th K , l e ads to

T h e r e l a t ion i s jus t Eq. (16) in the text . In con-

nection with the K,., prob lem a number of a u t h o r s

have ver i f ied that even if sub t r ac t ions are in t ro -

duced i11.f 's, when u s e of o t h e r inputs such as SU, x SU, spect ra l - funct ion s u m r u l e s , Ca l l an -Trc i - m a n re l a t ions , e t c , i s m a d e , r e s u l t s c l o s e to (C3)

a r e obtained.

(1963). 10 J. Bernstein and Ill. Gell-hIann (unpublished). See

also M . Gell-RIann: in Proceedings of the Tentlz An- nual International Roclzestev Corference oiz i-lgh- P n e v ~ y Pizysics, 1.960, edited by E . C. G . bXdarshan, J. I-I. Tinlot, and A . C . Melissinos (Interscience, New yorl<, 19601, p. 508.

"see revie~v article by I,. 1'1. Chounet, J, hl. Gaillard, and M. K. Gaillard. Phys. Rep. g, 201 (1972). This article gives a nunber of previous references.

1 2 ~ . I:. i\llaclrey, J, 37, RIcIiisic. D. M. Scott. and XT. \V. LVada, Phys. Rev. L l , 1590 (1968);=, 1520CEi (19693; D. \V, hIcKay, .J. hZ. L'ZcIiisic and W. W. Wada, ibirl. J8.1, 1609 (1969); J. C. Pati and I<. J. Sebastian, ibid. E, 2033 (1968).

'"ee, for example, P . A. Carruthers, intvorlz~ction to LTnitniry .Syrrlmetvy (Interscience, New York: 1966).

"particle Data Group, Phys. Lett. @, 1 (1974).