Vector meson pole dominance in Jψ decays into mesons

Volume 85B, number 1 PHYSICS LETTERS 30 July 1979

VECTOR MESON POLE DOMINANCE IN J /~ DECAYS INTO MESONS

Takashi OKAZAKI, Seiji OKUBO, Yfiichi HOSHINO and Kanji FUJII Department of Physics, Faculty of Sciences, Hokkaido Umversity, Sapporo 060, Japan

Received 28 July 1978 Revised manuscript received 24 April 1979

The vahdlty of the assumption that J/~ decays into mesons proceed via to, q~ and 00 poles followed by cascade decays is examined. F(J¢ --* Blr) Is well reproduced by using P(J/~p ~ #Tr), p(o~ ~ air), F(B ~ wu) and the B ~ t~lr hehcity structure. The structure of OZl-vmlatmg J/¢ -V ° transitions including the electromagnetic contribution is examined, and compared with the data on inclusive J/¢ decay

As various decay modes of J/ff (hereafter simply denoted as J) into ordinary hadrons are observed, it is interesting to consider the problem concerning the OZI [1 ] -violating interaction. In this note we analyse mesonic decays of J on the basis of the simple assump- tions that J decays via the 00, ~b and 0 0 poles (fig. la) followed by two-body cascade decays (fig. lb).

Firstly, the validity of our assumption is shown by analysing J ~ pTr and BTr. We obtain the strengths of the J - V 0 (V 0 = 60,4, p 0) transitions from various two- body decays of J, and investigate the structure o f OZI- violating J - V ° transitions. As we treat these transitions including the electromagnetic contribution, we inves-

J V ®

(a)

J

(b)

Fig. 1. (a) Vector-pole of J decays. - 0 - denotes OZI-violating J -V 0 transition. (b) Cascade diagram of J decays.

tigate the U-spin scalar property of these transitions. Throughout this note, the experimental data which

we use are quoted from ref. [2] and the references therein.

Strength of the J-00 transition. In this and the following sections, we consider the process J --} prr and J -~ Brr, to which the co-pole contributes. From Br(J -} prr)exp = (1.12 + 0.1 5)%, we get the effective J-00 transition strength 6 j~ as

[~Jw I2 = (0.89 + 0.12) X 10 -6 GeV 4 , (1)

where the effective hamiltoman describing the J-00 transition is written as

Hefr(J00 ) = * j~ J~00~ •

Here we u seg2pJ4 r r = 22.3 GeV -2 obtained from ['(co --} 3rr)exp = 8.99 MeV on the basis of the Gell- Mann-Sharp-Wargner (GSW) model [3] with finite p-width correction. (We use

Hint (00prr) = gtopu aa 00307 Pb e a ~ It ,

and obtain g2pn/47r = 19.9 GeV -2 without the finite p-width correction.)

In this connection, it deserves mention that the decay J ~ 37r allows us to examine the cascade decay dominance (or the complete p-pole saturation) rather directly, since J ~ pTr occurs really. Using the strength of the J - p - i t vertex determined by P(J ~ P~)exp, we find Br(J -~ 3~r)cal = (1.40 + 0.19) by calculating ]Tfi 12,

107


where

_ (+)_ ( - ) _ JerjX

Tfi ~X gjpngmr n L s _ m 2 + impF p

+ (p(+) _., p(0)) + ( p ( - ) _~ p (0))], J

where p(a) ]s the four-momentum ofTr a, s = (p(+) + p (-))2. Comparing the above value with Br(J ~ 31r)exp = (1.6 + 0.6)%, we see that we can explain the non- resonant part o f F(J -+ 370 (i.e., F(J ~ 37r) - F(J

/970) as mostly due to the interference among the three terms m Tfi, so that the cascade decay would be dominant.

Decay wMth and helicity structure o f J ~ BTr. The vector-axialvector-pseudoscalar (VAP) vertex is de- scribed by two independent Lorentz lnvariants, which correspond to two possible states o f relative angular momenta S and D:

T(VAP) = - G 1 e(V)ue(A)U + G2P,~p~ e(V)ue(A)v " (2)

In terms of r = m 2 G2/G 1 , the normalized helicity- zero amplitude for B ~ cow is written as

IFx= 0 (B -~ co~)l 2

(1 - rq2/mBEto) 2 = ,

2(mto/Ew) 2 + (1 - rq2/mBEto) 2 ' (3)

where q and Eto are the three-momentum energy and of co in the B rest frame. From the experimental data IFx= 0 (B ~ coTr)l 2 = 0.11 -+ 0.03 (or D/S amphtude ratio = 0.29 -+ 0.05 [4] ), we get r ~ 5. In the same way, we can write the decay width and the hehcity-zero amplitude of J ~ BTr in terms of r as follows:

16Jw 12 fJlr F(J ~ co ~ Brr) = 3 F(B ~ con),

(m2 - m 2 ) 2 fBTr (4)

I F x = o O -~ co - , Blr ) l 2

(1 - rmjp2/m2 EB) 2 =

2(mB/EB) 2 + (1 -- rmjp2/m2 EB) 2

(5)

where p and E B are the three-momentum and energy

of B in the J rest frame,

=2m 2p { + P2 - r ~ 2 , m +r2 p4},

3m 6

(6)

fB~r - q 1+ - r ~ ~.__ q +r 2 2 3m 2 m~orn B 3m2m2 B "

q and E ~ are the momentum and energy of co in the B rest frame.

From the above formulae and the value 16j,~l 2 in eq. (1), the observed decay width of J ~ Brr is well reproduced by using I ' (B ~ coTr)exp and the value r determined from the observed helicity-zero amplitude (see fig. 2a). This shows the validity of the co-pole dominance assumption and we predict IFx=0(J ~ BTr) l 2

0.99, as shown in fig. 2b + i.

t l After complet ion of this work, we noticed the first experimental data on the hehcity structure of J ~ Bn [5 ], which

seem to show a rather large fraction of helicity-one state

0 , 6

o,5

0,4

v L..

42

0j

i i i i i ' t

(a)

.'i

0,1 0,2 03 0,4 0.5 0,6 0,7 IF)..,(B ~(,.)'n') 12

~_ 1 , 0 ~

m 0 , 5 1 - I'

-~o LL"

0

I f i l l

(b)

l i t

5 r

Fig. 2. (a) F(J --~ BTr) versus IFh=0(B ~ tmr)l 2 as a functaon of r. (b) IFk=0(J ---, Blr) 12 versus r

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Strength of the j _ p 0 transition. The j _ p 0 transition is caused by the electromagnetic interaction. Calculat- ing its contribution from a diagram with a one-photon intermediate s ta te , ~jp can be expressed in terms of the V- ' y coupling at q2 = m 2 as follows:

~jp e2m2fj7 2 2 2 = (mj)mpfp~/(mj) m/-2, (7)

where fv~ (q 2) is defined as <0 IJe~m IV) = m2fv.r(q 2) × e(V)V. Setting fm(m 2) = fp~(m 2) and using the experimental data on p, J ~ eE, p ~ mr and A 2 ~ pTr, we find Br(J ~ lr+rr - ) ~ 0.01% and Br(J ~ A27r ) ~2.8%. The latter conflicts seriously with the experimental upper limit of Br(J ~ A21r ) < 0.43%, which indicates a large mass dependence in the D-wave #A27r coupling and/or in the p-~, coupling.

In table 1, we summarize the J - V 0 transition strengths obtained from non-strange modes of J ~ PP, PV and PT. Here and in the following, we assume the SU(3) symmetry and mass independence of the strong couplings. (In the final section, we will comment on the mass dependence.) Concerning J ~ T V ( s u c h as J ~ A 2 P ,

f ~ , f ' $ , fp, etc.), there appears much complexity due to the fact that five Lorentz invariants may occur in the VTV vertex. Nevertheless, one can see that the experimental ratio P(J ~ A2P)/I-'(J ~ fw) ~ 3 is well reproduced by the nonet relation, because these two proces- ses are almost the same kinematically.

J decay into a strange-meson pair. Next, we consider J decays into a strange-meson pair, to which all of ~o,

and pO poles contribute. We define the following factor D, which appears in the expressions of the decay widths:

{1 + 2 2 2 2 5jO/2 m 2 - mto (~jp mj - mo~ D =

~ jto +/3N/~ . ' ~ -- - ~ - ~-7-j~j ' (8) t X m j - - m p m j - m e

where a and/3 are sign factors; in the case of KI~ K 'K* and KI(** (KI(* and K*I(** ) final states,/3 = - 1 (+1), and for the neutral (changed) modes, a = - 1 (+1). In deriving eq. (8), the nonet relations of strong couphng were used. Thus, we obtain the following relations:

l"(J ~ K+K - * * ) _ l"(J ~ K+*K - * ) _ r ( J "-~ K+K - )

r ( J ~ K°K °**) r ( J ~ K°*ff. °*) r (J-~ KsKL) '

r ( J -~ K+*K -**) = r ( J -~ K+K -*) (9)

r ( J -~ K °* K °**) r ( J -~ K°ff. °*) "

In order to go further, we suppose that the J - V 0 transitions are U-spin scalar; it contains SU(3)-octet and -singlet parts of electromagnetic and strong origin. Thus, we can re la te t~jq~/t~jw w i t h ~Jp/~Jto as follows:

~ J J ~ i J w -- 2-1 /2( 1 - ~Jp /~Jco) , (10)

so that the D ' s are expressed by only one parameter ~Jp/~Jo~. From various experimental data and the value (1), ranges of fiJ#/~Jto are determined as shown in fig. 3, together with the results from non-strange decay

I ' ' ' I ' ' ' ' I ' ' ' '

Table 1 The J - V o transition strengths determined from non-strange modes. Here we use the mixing angles 0 V = 35.3 ° and Op = - 1 0 ° .

Decay Branching raUo I~ jV I 2 (GeV 4) mode

la jpl 2

J ~ , r + n - (1 .0+-0 .7)× 10 -4a) (0.73 +-0.51) X 10 -6 (1.6 +- 1.6) X 10 "4b) (1.2 -+ 1.2) X 10 -6

J ~ A ~ l r - <0.43 X 10 -2 <0.13 × 10 -6

15J~ol 2 J --~ plr (1.12-+ 0.15) X 10 -2 (0.89 -+ 0.12) X 10 -6

2 16Jt#l

J ~ ~0 (0.10 -* 0.06) X 10 -2 (0.35 -+ 0.27)X 10 -6 J---~Or/' <0.13 X X 10 -2 <0.66 X 10 -6

a) DASP. b )SLAC-LBL.

0,3

o

H

I

I-,,-4

I-*'4

• •

H

Inclusive

~"n-(SLAC- LBL)

IK~K" i

I~K-(SLAC-LBL) KK(DASP)

• I

I# , , , I , a I , J ~ ,

o lo

sj,/sj Fig. 3. The ratio a j p / S j w determined from various decay modes with the assumption of U-spin scalar J -V o transition and the nonet relation in VMM' coupling. From the experimental upper limit on r ( J ~ KLK S) we get I/ijp/~Sjtol < 35.

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modes. Although there are large uncertainties in the experimental data, especially in J ~ n+n - and K+K - , we may set 8 jp /6j t o ~ 0.3 with a global consistency, ex- cept for J ~ K+K - * .

The following remark should be added on the large discrepancy in J ~ K+K - * . In our model, we may at- tribute this discrepancy to certain kinds of symmetry breaking in the J - V 0 transitions or/and VVP couplings. It seems difficult to remove the discrepancy even if we introduce a violation o f U-spin invariance in the J - V 0 transitions (i.e. abandon relation (10)), while a nonet symmetry breaking in the VVP coupling gives a satis- factory result. The latter comes from the fact that a nonet symmetry breaking in the VVP coupling sensi- tively changes the t~jp/Sjw dependence of the factor D in the case o f J ~ K+K - * ,2 . Then, the allowed

+ - - * r a n g e oftSjp/~jt o determined from P(J-+ K K )exp is easily shifted due to such a symmetry breaking. This symmetry breaking is expected from the experimental ratio of P(K 0* ~ K07)/I '(co -'- rr07) as indicated by Okubo [6]. A similar situation occurs in the case of J

KLK S and K0ff, 0 .* , but the experimental data on them are not enoughto discuss these points at present.

Concerning J ~ KKn, we analyse this process accord- ing to the same procedure as we did for J ~ 3rr and pzr. The observed process J ~ KsK- n + proceeds through K+. , ~ 0 . and " p - " poles:

J -~ KOKO* ~ K - K +* ~n+, ,p - , ,

~ K - r t + ' ~K07r +' [ + K - K 0

with finite width correction to the K* and P propaga- tors, we get Br(J ~ KsK- r t +) = (0.23 -+ 0.94)% from observed rates o f J ~ K+*K - , K°*K0 and pTr. Com- paring these values with the experimental data (0.26

,2 The factor D defined In eq. (8) for J ~ KK* can be written by the remaining W P couplings as follows'

D=I{I+m~-mL gOKK*I m] - rn~ gtoKK*)

j - m ~ gtoKK* m~ m~ g-~-*JSJcoJ "

In the nonet symmetry hm]t (gpK+K-,[gwK+K-, = gOK+K-*/',/2gtoK+K-* = 1), the coefficmnt of 8jp/Sjt o almost vanishes.

-+ 0.07)%, we can conclude that the cascade decay is dominant.

Summary and discussion. We have shown that the decay widths of J ~ Bn and pzr are well understood by co-pole dominance, and that the three-body decays of J into 3n and K s K - n + proceed mainly via cascade decays. This presents us with clear evidence o f the validity of our simple assumpnon.

We have analysed various decay modes of J on the assumption of U-spin scalar J - V 0 transitions and found that a global consistency is obtained for 8 jp /Sj t o ~ 0.3. From analysis of the reclusive data on J ~ hadrons, one obtains P(J ~ 7 ~ hadrons)/P(J -~ hadrons)tot ~ 0.20 + 0.04 [7]. This implies [Sjp/8jw [ ~ 0.45 -+ 0.05, which may be consistent with the above value ~ 0.3. Here it should be noticed that, m determining the value of 6jp/Sjw with a global consistency among mesonic J decays, the experimental upper limits of P(J ~ A2n ) and U(J ~ K+K -**) give a strong restrictxon. (Remem- ber that we have used the value of V °TP coupling determined from the physmal A 2 ~ pzt.) If the VOTp coupling (which leads to purely D-wave decays of J ~ PT) decreases in going from q2 = m 2 o to m 2, as suggested by Kramer et al. [8] , one obtains a larger value of 8 jp / 8jr o, which is consistent with the existing data including the inclusive ones.

From the analysis o f J ~ K+K - * , it is suggested that there exists some nonet symmetry breaking in the VKK* coupling. More precise experimental informa- tion on the second-order electromagnetic decays and strange decay modes will test our statements.

References

[1] S. Okubo, Phys. Lett 5 (1963) 165; G. Zweig, CERN preprint TH412 (1964), unpubhshed; J. lizuka, K. Okada and O. Shito, Prog. Theor. Phys. 35 (1966) 1061.

[2] G.J. Feldman and M.L Perl, Phys. Rep. 33 (1977) no. 5. [3] M. Gell-Mann, D. Sharp and W.G. Wargner, Phys. Rev. Lett.

8 (1962) 261. [4] Parncle Data Group, Review of Particle Propertms, Phys.

Lett. 75B (1978) 1. [5] J. Burmester et al., Phys. Lett. 72B (1977) 135. [6] S. Okubo, Phys. Rev. Lett. 36 (1976) 117. [7] M. Boyarski et al., Phys. Rev. Lett. 34 (1975) 1357. [8] G. Kramer, J.L. Uretsky and T.F. Walsh, Phys. Rev. D3

(1971) 719.

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Vector meson pole dominance in Jψ decays into mesons