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Volume 133B, number 1,2 PHYSICS LETTERS 8 December 1983

EVIDENCE FOR CHARMED-STRANGE EXOTIC MESONS? ~r

M. SUZUKI Department of Physics and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA

and

S.F. TUAN Department of Physics and Astronomy, University of Hawaii at Manoa, Honolulu, HI 96822, USA

Received 13 September 1983

We suggest that the 1970 MeV state recently discovered in ee - annihilation at CESR is the F meson and the old 2020 MeV state is the (cgqCt)l= O state (q = u, d) ofJ P = 0 . SU(3) analysis indicates a rich spectrum of charmed four-quark states. Semi-quantitative arguments are presented for the reason why the 2020 MeV state decays dominantly through weak inter- actions.

The discovery of a new state with narrow width at 1970 MeV decaying into 1r + [1] has generated a big puzzle in the charmed-strange meson spectroscopy. Is this for instance the F + (JP = 0- of c~) meson? If so, what is the particle at 2020 MeV that was previously reported by a few experiments [2] and identified with the F+? The reported width of the 1970 MeV state is consistent with experimental resolution and the life- time of the 2020 MeV state is characteristic of weak decays. It is therefore very unlikely that either of these states is one of the glueball states, (qqg) states, or noncharmed four-quark states. In this letter we as- sume that the above states are really two distinct par- ticle states and address ourselves to this puzzle by iden- tifying the 1970 MeV state with F + and the 2020 MeV state with the isosingle c~qffl state. We work out the flavor SU(3) analysis of states, argue that weak decay can dominate for the 2020 MeV state, and point out that there should be two D-like four-quark doublets in the 2 GeV region, one of which may decay into Drr in S wave.

Systematics of four-quark states have been discussed by many authors [3,4], notably by Jaffe. Importance of charmed strange mesons was pointed out by Lipkin

This work was supported in part by the National Science Foundation under Grant No. PHY-8118547 and by the US Department of Energy under Contract DE-AM03-76SF00235.

0.031-9163/83/$ 03.00 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

[5]. We first describe the systematics of charmed four- quark states in the flavor SU(3) symmetry. According to empirical rules of light quark dynamics, two quarks (antiquarks) tend to be attracted more strongly in the color 3* (3) channel than in 6 (6*). The spin-spin force aligns quark spins in antiparallel configuration. Therefore, it is most likely that the lowest four-quark states of cq61~l, where q = u, d or s, are formed in the flavor SU(3) representations that result from 3 3, namely, 3* and 6 * ~. The quark contents of these states are given in fig. 1 and table 1. These states have JP = 0 +. As to which of 3* and 6 is the lighter multi- plet, this depends on the quark potential; A general guiding principle is that a multiplet of lower SU(3) representation is less heavy than that of the higher one. Accepting this rule the lowest four-quark state with the same charm and strangeness as the F + is the 1 = 0 state F'x ,2 of 3", a state which was called crypto-exotic by Jaffe [3]. The 6 representation is exotic and contains the 1 = 1 state F'I with C = S = +1 and the 1= 0 state F S with C= +1 and S = -1 . The Gell-Mann-Okubo mass formula for 6 is linear like m I = m(6) + Am I. However, the quark model further imposes that ml= 1

,1 The SU(3) flavor decomposition of cqqff does have pieces from 3 6", which are expected to appear as higher mass multiplets.

2 We follow the particle notations of ref. [5 ].

125

Volume 133B, number 1,2 PHYSICS LETTERS 8 December 1983

-1

6

V ......... ~ : :- / mix

I I I

- i 0 1

S

I /

. . . . . . . . ~" 0

-!

t I

-1/2 1/2

I3 I 3

Fig. 1.6 and 3* states of cq'~cl. See table 1 for quark content.

= ml= 0 since both /= 1 and 0 states contain one s or quark. Therefore, we expect Am ~- 0 and degeneracy

of 6 states to the first order in SU(3) breaking. It _ 1 3* should be noted that the I - 2 states of and 6 mix

with each other through SU(3) breaking to generate two mass eigenstates. The mass eigenstates are ex- pressed as

D '= D(6) cos 0 + D(3*) sin 0,

= - ~(6) sin 0 + ~(3" ) cos 0, (1)

1 where D(6) and D(3*) stand for the 1 - ~ states of 6 and 3", respectively, and 0 is the mixing angle related to the 6 - 3* transition mass m(6-3" ) by

tan 20 = 2m(6- 3" ) / [m(6) - m(3*)]. (2)

The mass eigenvalues are given as *3

m~, + rn~ = m~ l + ~ mF x ,

rn~, - m~ = (rn~l - m~. x ) [1 + tan220] 1/2. (3)

The ideal mixing is realized with 0 = 45 such that D' = cs~61 and D = {cqCtq~l- 1/2 like ~ and ~, where q = u or d, and these states are likely candidates for the mass eigenstates in the naive quark model. It is interesting to note here that the ideal mixing 0 = 45 forces degeneracy of 6 and 3", as is evident from eq. (2). In this limit we predict masses of D and D' as

= ~ - = ~ +Amsq, (4) rn~ mFx Amsq , m~, mFx

where Amsq = m s - md, u = mF - mD ,0. For values of 0 other than 45 , m(6) is not necessarily equal to m (3") and. D' contains a (cq~]61)l= 1/2 component.

It is natural to assume that fix (-csqcl) is no lighter than F+(cg) since addition of (q(l)l=0 should cost some positive energy;,Therefore, F is assigned to the 1970 MeV state and F x is assigned to the 2020 MeV state. The absence of a peak around 2020-2050 MeV [6] in the Cp0 invariant mass plot does not allow us to iden- tify the 2020 MeV state with F'I" With our assignment, the F+-D + mass difference comes out on the smaller side (1970 MeV- 1869 MeV = 101 MeV) as compared with A -N mass difference and Y~-A mass difference, but it is comparable with the K* -p mass difference (123 MeV) and the K* (2)-A2 mass difference (116 MeV). With this value of Amsd = 101 MeV, eq. (4) leads to

mg+ = 1915 MeV, ml~o = 1919 MeV,

Table 1

if7 + = (1/21/2) cu (ds - "sd)

~/+ = (1/2) cu fu -fis-) + (1/2) cd (d's -~,d)

F'~ = (1/21/2) cd6u - fi's)

5(6, 3*) = (1/2) cs(d'~ - ~d) + (1/2) cu(fid - dt~)

5 (6, 3*) = (1/2) cs(st~ - fis-) + (1/2) cd(t~d - dt~)

rag,+ = 2125 MeV, rns~o = 2121 MeV. (5)

A problem subject to more debate is whether the added q61 pair can cost only 50 MeV (= 2020 MeV-1970 MeV). We do not find a convincing argument to the contrary, so we proceed by assuming that this is pos- sible.

Next we discuss the decays of these particle states. The most important problem is obviously whether the F~ x lifetime can be as long as 2 X 10 -13 s or not. F'x can decay into F + by emitting two ")" since one ~' emis-

v~ = (1/21/2) cs(6~ - ~6)

Fx = (1/2) cuG~ - fi~) - (1/2) cd(d~ -~d)

.3 In the naive quark model, we expect m~(3. ) = m~x by s quark counting, so m (3*) has been replaced by m]~ x in eq. (3).

126

Volume 133B, number 1,2 PHYSICS LETTERS 8 December 1983

sion is forbidden by the 0 -0 transition rule. To be consistent with the observed lifetime, the radiative de- cay F x ~ F + + 77 has to be suppressed enough to be dominated by the weak decay of the c quark in F'x" One might make a naive estimate in analogy with 2S

IS + 3'3' in hydrogen, which leads to the expression I'F.rv ~- (1/121r) a2(A/M)aA [7] where A = m~" x - m F = 50 MeV and 1/M is the size of the spread of light quarks. With M -~ constituent mass of u, d, this formu- la would give

FF~/Fc-- , suc~ ~ ( 327r2a2 A/3 GF 2 mc 5 ) (A/mq)4

= 1 102. (6)

However, this is probably a gross overestimate because the formula applies to the 23' emission of the type shown in fig. 2a, not to the process of our present in- terest, as depicted in fig. 2b, where a light quark pair must annihilate to produce two 7's. Estimate of such a matrix element needs a detailed knowledge of wave-

C '~

g ,,

q

Ca

C

g ,c

(b)

q ~, y

(c)

functions of light quarks inside F'x. One way to ap- proach this problem is to estimate through

Fx -+ F + "r/" (7 ) t-,77'

Current algebra sets a constraint that the matrix ele- ment should vanish at p~ ~ O. Evaluating the matrix element including this condition, one finds an order of magnitude estimate

YFv,r "~ (1/192rr3)(e2/327r2frr)2 A9/f~rmn 4 (8) leading to

I 'F~v/I 'c~ suci ~ (~2/96rt2) A9/GFmcf~r2 5 4mn4

= 1 10 -5. (9)

The large reduction of I-'F77 in eq. (9) relative to I'F.r. r in (6) is due to the current algebra zero characteristic of light 0 - states and to (1/32rr2) 2 in 77 -+ 73'. Another possible decay mechanism is to emit 23' from ~ quark with a qc] pair annihilating into two gluons that even- tually are absorbed by c and/or g. (See fig. 2c.) The suppression of this process relative to that of fig. 2a is the s quark charge to the fourth power ( -1 /3) 4, replace ment of mq in eq. (6) by constituent s quark mass, and whatever factor that costs to annihilate q61 through two gluons. Taking the last factor rather arbitrarily to be 1/10, we find a suppression factor o f ( l /3 ) 4 (300/ 500) 4 X (1/10) -~ 1.6 10 -4, which is sufficient to reduce the FF.r.r/I'c~ sud ratio to less then unity. The estimate of the decay rate I'F.r. r is crucial to our pres- ent assignment of particle states and improvement in accuracy of estimate is in pr.~ogress. It is quite possible that the radiative decay.,of F x is strongly suppressed and the main decay of F x is through the weak interac- tions. In the weak decays, final states ofF" x consist of three quarks and three antiquarks, which, some people argue, form more