The spectra of qq states and glueballs

ELSEVIER

Nuclear Physics B (Proc.

The Spectra of

qij

Suppl.)

96 (2001) 218-224

www.elsevier.nvlocate/npe

states and glueballs

D.V. Bugga “Queen Mary and Westfield College, Mile End Rd. London El 4NS, UK Data states

from LEAR (J

of s =

Gauge

1.

The

pp + final states with I = 0, C = +l reveal s-channel

= 0 to 5) expected M’

trajectories: decays:

on

v.

in the mass range

radial

excitation

&3(1500),

f0(2105),

this suggests calculations

spectrum

London

glueball

number. fi(1980)

character.

of Morningstar

1920-2400

In addition and tj(2190). Their

and Peardon

MeV/c* there

They

are four extra

All have exotic

states

properties

mass ratios agree within errors for the lowest

of I = 0 C = +1 qtj states

The Crystal Barrel experiment at LEAR has produced data at nine I? momenta on pp + neutrals wit,h very high statistics: a t(otal of - 20 million kinematically fitted events. Twenty final states have been analysed to produce physics. Exa.mples of these final states are #7r”, 3~‘: v~“‘K’, I~T’olrOirO.Ld?P (w + 7-i’-,). Partial wave analyses of many of these final states have been completed [l-7]. Here, at,tention will be concentrated on I = O> c’ = +1 final states for ttio reasons. Firstly, t’he spectrum of observed states is most, complete and provides a valuable basis for comparison with models of confinement. Presently none of the models provide accurate agreement with experiment. Secondly, the I = 0, C = $1 quantum numbers are those which allow glueballs. Four states with unusual properties fit nicely with the latest Lattice Gauge predictions for glueballs. The main results of this paper are to be found in Table 1 which compares mass ratios of glueball candidates with these calculations, and in Fig. 2, t,he spectrum of observed non-strange qQ states. The channels for which data are available with t,hese quantum numbers are:

resonances

lie on parallel

corresponding straight-line

which

fail to fit on to these

and appear

(typcially

to all qq trajectories

in J/3

3~4%) with recent

radiative Lattice

four glueballs.

+

7r-lr+

+

q7’”

+

317.

I

0

0

(5) (6) (7)

For the 7r- TT+ final state, the data come from three earlier experiments [8-lo]. The steps in the analysis were as follows. Firstly, the r]nOnO data were analysed at the nine available momenta into the channels a%( 1320) no, j~(1270)7], no(980)r0, fo(980)r”, fo(1500)n’ and ~7. The first two of these final states are dominant. From production angular distributions and decay angular distributions of a? and fz, it is found that partial waves up to Jp = 4++ are required. The well-known f4(2050) makes a large and clearly identified contribution shown on Fig. 1. It acts as a powerful interferometer. The data determine relative phases of other partial waves with respect to this reference wave. Peaks are observed in these other partial waves. However, relative to 4+, phases vary little with jj beam momentum. The conclusion is that there must be resonances in many partial waves, possibly all of them. In Refs. [l], ten resonances were required to fit the intensities and phases observed at individual momenta. The next step was that a combined analysis was made of R’K’, J/V, ~7’ and r-r+ final states [‘2]. In the first three channels, only I = 0, Jp = O+, 2+ and 4+ are allowed; in T-A+, I = 1 Jp odd states are also present. On the basis of experi-

0920-5632/01/$ - see front matter 8 2001 Elsevier Science B.V. All rights reserved PI1 SO920-5632(01)01133-l

D. c! Bugg/Nuclear

s 10 6

‘Z

P E D

5

Physics B (Proc. Suppl.) 96 (2001) 218-224

0)

ence from &)?T’, these data were fitted directly in terms of s-channel resonances. This prescription has the virtue of satisfying the important constraint of analyticity, since a Breit-Wigner amplitude is an analytic function. In low partial waves, backgrounds were introduced where necessary as constants or poles below the pp threshold. These backgrounds may be due to lower resonances or may parametrise t-channel exchanges. Again the functions are analytic. The ?y”, T] and q’ are members of an SU(3) nonet. Therefore n”rro, 1777and r~q’should be connected by SU(3) relations. In practice, there is a minor breaking of SU(3) due to form factors and small phase differences between channels. However, the overall picture is that all states except one are compatible with being (uii + dd) states, with only small SS admixtures; details are to be found in Ref. [2]. The one glaring exception, discussed below, is fs(2105), where a large violation of SU(3) is observed. The final step in the analysis, completed recently [3], is to combine the analysis of these twobody channels with r7n0no and r]‘nOnO data. In this step, the latter channels are fitted at all beam momenta simultaneously to s-channel resonances, together with all the twobody data. This fit is highly constrained. It succeeds in locating all the expected non-strange qtj resonances in the available mass range, 1920-2410 MeV. The resulting spectrum is shown in Fig. 2. It has very simple characteristics. All resonances lie close to straight-line trajectories of s = M2 v. excitation number, with an average slope of (1.143 f 0.013) GeV2 per excitation. Fig. 2 also includes the I = 1, Jp = 3- trajectory, which is particularly well determined. This pattern of resonances will hopefully be useful input in attempts to understand confinement. It will be used here as a basis for identifying a number of ‘extra’ states, which fail to fit into this pattern. Those states which have abnormal characteristics and which fail to fit into the simple pattern of Fig. 2 are f0(1500), f0(2105), f2(1980), v(2190) and ~(1860). The first four have the quantum numbers predicted for low-lying glueballs. The TJQ(1860) is a candidate for a 2-+ hybrid expected

ll!!Yl ,, . ,s’I

,,,i

t

_+__I-

2

I’

2.2

2.4

I.4 (GeV)

2

2.2

2.4

Figure 1. Contributions to (a) f2(1270)~, (b) ns(1320)n from 3F4 (black squares) and 3H4 (open triangles)

0)

b)

”

”

6 23

4

2040

3

1770

3PO 5 i_

:L

Cl

n

0

1

2

3

4

d)

Figure 2. Plots of M2 v. radial excitation number, 71. Straight line trajectories are drawn in all cases with a slope of 1.143 GeV2; numerical values give masses in MeV.

”

219

D. F Bugg/Nuclear Physics B (Proc. Suppl.) 96 (2001) 218-224

220

near this mass [II]. perimental evidence q(2190). 2. Glueball

1

Ii5

15

175

2

215

25

275

5

3 2'

250

Q

200

3 al

(5 u)

150

0’

9 >

100

2 cs

50

0 1

1.5 MI,

2

2.5

3

G@J/c2)

Figure 3. (a) The 17~mass distribution from E760 data for pp + vqnO at 3.0 GeV; (b) the 4x mass distribution from DM2 data for J/\E -+ y(47r).

I shall now review the exfor f0(2105), fi( 1980) and

Candidates

2.1. fo(2105) This resonance has been observed in several sets of data. The quantum numbers are found to be Jpc = O++ in three separate determinations. It appears very clearly as a peak in E760 data on j@ -+ 777~‘. However, in this experiment angular coverage was not sufficient to allow a Jp determination. Three peaks are visible in the 7717 spectrum, Fig. 3(a), at 1500,175O and 2105 MeV. Similar peaks are visible with lower statistics in Mark III and DM2 data on J/q -+ 7(4n), Fig. 3(b). An amplitude analysis of the Mark III data [12] showed that all three peaks have Jpc = O++. Next, this resonance appeared in Crystal Barrel data in flight for pp + (vq)lr” [13]. The angular analysis was compatible with Jp = O+ or 2+, but t$he energy dependence of the production cross section required O+. More recently, the amplitude analysis of pp + qq [14] has demonstrated the appearance of a strong Ot peak with mass and width consistent within f10 MeV with E760 values. A remarkable feature is that fo(2105) decays much less strongly to n”?yo than expected for a simple qq state. The r] has wave function (rj >=

0.8?

- 0.6.9s.

Hence the branching ratio ~“~o/~~ is predicted to be 1/(O.8)4 = 2.45. Experimentally, this ratio is 0.71 f 0.17. If one expresses this result in terms of mixing between non-strange and strange components, lfo(2105)

>= cos Qy

+ sin ass,

(9)

the mixing angle 0 is (58 f 5)‘. This is far from the value O” for a non-strange meson. For a glueball, @ = 35.6’. 2.2. f2( 1980) This resonance central production

made data

its first appearance in on 47r final states [15].

D. I! Bugg/Nuclear

Physics B (Proc. Suppl.)

More recent WA102 data [16], shown in Fig. 4, show a well defined peak in fs(1270)nn with I%[= 1989- 1995 MeV, r = 436-448 MeV in two different charge combinations. The WA102 group has studied the dependence of many produced resonances on dPT, the difference in transverse momentum vectors of the two exchanged particles. All undisputed qq states such as 17,$ and fl(l285) are suppressed as dPT -+ 0. Glueball ca,ndidates instead appear peaked at dPT = 0; the f2(1980), like fo(l500), appears at smalldPT. Secondly, the azimuthal angle 4 between the PT vectors of the two outgoing protons is found to be very different for well-known qq states such as fi( 1270) and for glueball candidates fo(1500) and fz(1980) [17]. Th is empirical observation is illustrated in Fig. 5, but is not yet fully understood. The fz(1980) appears in two sets of Crystal Barrel data. It was first observed in pp + (~~)~‘O in flight [13]. The decay angular distribution at several beam momenta shows strongly nonisot)ropic decays, clearly visible above an qq mass of - 1550 MeV. It requires spin 2 in vq. It cannot be fitted as f~(2050) or by a combination of fz( 1565) and fi( 1920). The mass shows a clear optimum at 1980 & 40 MeV and the width at 500 5 100 MeV, in close agreement with WA102 dat,a. Secondly, the combined analysis of data on final states nor’, 717,qq’ and x-r+ [3] requires a broad component with similar mass and width, in addition to the four qrj states of Fig. 2 at 1934, 2001, 2240 and 2293 MeV. The fitted mixing angle is @ = 23.5 f 3.5”, giving an sS intensity midway between that of a non-strange qq state and a glueball. There are further tentative observations of fi(1980) in BES data on J/rk + y(47r), again in fi(1270)a, [18] and in y(K*K*) [19]. In the mass range 2020-2300 MeV, Etkin et al. [20] have observed a ?T?T4 ~~ signal 100 times larger than expected from small violations of the OZI rule. The interpretation I place on these data is that peaks at 2150 and 2300 MeV are ss 3P2 and 3F~ partners of fz(1934) and fz(2001) of Fig. 2. If a broad 2+ glueball lies in this mass range, it accounts naturally for mixing between (UU+ d2) and ss states, hence significant rn -+ 44.

96 (2001) 218-224

221

M (n+fn+n-)

GeV

The Jp = 2+ fz(1270)nr mass disFigure 4. tribution from WA102 data for central production of lr+7r-x+7r-.

0.3

0.3

0.25

0.25

0.2 E n 0 r-J

0.2

o.is

0.15

0.1

01

0.05

0.05

\

p

Deg

Figure 5. The 4 distribution for WA102 data on central production of (a) f0(1370), (b) f0(1500), (c)fi(1270) and (d)f2(1980).

D. I! Bugg/Nuclear

222

Physics

B (Proc.

Suppl.)

96 (2001)

218-224

q(2190) Data on J/9 + y(~.rr+x-) and r(K1?n) cont,ain the well-known narrow ~(1440). Its produc-

This term is required

t,ion in J/Q radiative decays led to early speculat#ion that it is a O- glueball. However, its mass now appears to be much lower than that predicted for a. glueball. BES data on J/Q -+ r(K’rc7r) require coupling to IS*(980)1( stronger than that to rl~~ [21]. BES also observe radiative decays to

the effect of (A4: - s) in the denominator of the Breit-Wigner amplitude. For a glueball, one expects the coupling constants to be in the ratio

2.3.

y(&y) stronger than to y(py) [22]. Both sets of observations suggest that it may be interpreted as the ss radial excitation of q/(958), though its strong

production

suggests

in J/q

some mixing

radiative

also

with a glueball.

The early analyses however to a very broad contribution 17(1440). radiative

decays

paid no attention lying under the

This broad signal appears also in J/S decays to li’l?r, pp, ww, K*k* and +$.

A combined

analysis of these channels

[23] assigns

t)o this continuum a dominantly O- contribution. What has emerged is that this continuum may be fit,ted as a very broad resonance

with full width

at, half maximum 860 MeV. Considerable care is required, however, in including the required sdependence t.pcl as

into the width.

The amplitude

is fit-

(IO) where rtot is the sum of the decay widths to channels pp, ww, K*I?‘, 44, qmr and Kl?w; each IY has the s-dependence given by the phase space for the final state (folding in the width of final st.at#es like p and K*).

Fig.

6 shows these widths

and Fig. 7 the resulting branching ratios for each channel as a function of mass. Let us take vn~ as a.11example. The branching ratio IX lf121?(~lrn). For low s values, the ~~7r channel

dominates

and

t.he branching fraction of r]rn rises with s. However, at N 1550 MeV, the rapid increase in rpp denominator pulls a,nd Tww in the Breit-Wigner t,he branching

fraction

of 17”” down.

Likewise,

at

N 1800 MeV, the rapid rise in the Ii’*I(* width suppresses the branching fractions of pp and ww. An important element in equn. (10) is a dispersive correction m(s) to the mass:

M2-s

m(s) = ~

7r

s

ClS'

MOrtotal(4

(s’ - s)(M,2 - s’) .

(11)

to make equn.

(10) analytic.

It is zero for a Breit-Wigner resonance of constant width, but plays a strong role here, overwhelming

2 2 . 2 9 pp . h%w : g,.R*

2 -3:1:4:1. : 944 -

(12)

These values follow from counting the nine possible quark combinations for a flavour singlet ( uu + dd+ ss).

Experimentally,

the observed

branching

ratios agree with flavour bindness within experimental errors, which are f30% for pp, ww and

Ii* I?*, but 100% for 44. There is a further feature broad object is a glueball.

suggesting that Close, Ferrar

this and

Li [24] have shown that branching fractions may be predicted for production of O+, O- and 2+ glueballs in J/Q radiative decays. For fo( 1500) and 7(2190), these sum rules are obeyed within the experimental errors of - *25%. However, for f2(1980)

and fo(2100), the observed signals This may be beare smaller than predicted. cause some decay modes have not yet been observed in J/Q radiative decays, e.g. fo(2105) and f22(19W

3.

+

777).

Remarks The

on the Spectra

straight

line trajectories

of Fig.

2 may

not be exact. Many local perturbations to masses are possible, for example (a) spin-orbit and tensor splitting,

which may vary with s, (b) mixing

with glueballs or hybrids, (c) the effects of nearby thresholds, (d) 1eve1 repulsion between 2+ states. Furthermore, trajectories silon 3S1 states certainly slope

against

s, perhaps

are less relativistic Recent

Lattice

of charmonium and upexhibit a quite different because

these

and have smaller Gauge

calculations

systems

radii. of Morn-

ingstar and Peardon [25] in the quenched approximation provide an interesting comparison with the glueball candidates discussed above. There is a scale error for predictions of glueball masses of *lo - 15%, according to how this is normalised. However, mass ratios are predicted much more accurately. Table 1 [26] shows predicted mass ra-

D. K Bugg/Nuclear

Physics

B (Proc. Suppl.)

96 (2001) 218-224

223

Figure 6. Widths l?(s) for (A) ~?T?T, (B) pp, (C) ww (dashed), (D) K*lT’*, (E) q!$, (F) KIT0 and (G) total.

1

I.5

2

2.5

Mass (Gr V)

Mass

n .Mass (Gr b’/

Figure 7. Branching fractions for J/\Ir + yX, where X is (a) QK~, (b) pp, (c) ww, (d) K*ki’*, (e) q5q5and (f) total. Curves show the fit with 7j(2190).

(GeV)

11. V Bugg/Nuclecrr

224

t.ios compared

Physics

with those for fo(1500),

fo(2105),

f~(lY80) and v(2190) (whose masses were all det.rrmined beforethe predictions). There is remarkable agreement. st.at,es ha.ve significant

This suggests that these glueball content, though

observpcl decays require some mixing with neighbouring qq states. An understanding of this mixing and of decay modes in now the challenge.

As

yet,. La.ttice QCD provides little guide to mixing or decra.ys, because of the quenched approximat.ion.

Ratio hf(2++)/M(O++)

M(O-+)/M(o++) n/r(o*++)/M(o++) M(O-+)/A,I(2++) Ta.ble 1

Prediction

Experiment

1.39(4)

1.32(3)

1.50(4) 1.54(11) 1.081(12)

1.46(3) 1.40(2) 1.043(36)

Glueball mass ratios predicted by Ref. [25], compa.red with experimental candidates described in the t,est; errors are in parentheses.

B (Proc.

Suppl.)

96 (2001)

218-224

10. A.A. Carter et al., Phys. Lett. B67 (1977) 117. 11. A.V. Anisovich et al., Phys. Lett. B477 (2000) 19. 12. D.V. Bugg et al., Phys. Lett. B353 (1995) 378. 13. A.V. Anisovich et al., Phys. Lett. B449 (1999) 145. 14. A.V. Anisovich et al., Phys. Lett. B468 (1999) 309. 15. F. Antinori 589. 16. D. Barberis 440.

et al., Phys.

Lett.

B353

(1995)

et al., Phys.

Lett.

B471

(2000)

17. F.E. Close, A. Kirk and G. Schuler, Lett. B477 (2000) 13.

Phys.

18. J.Z. Bai et al., Phys. Lett. B472 (2000) 207. 19. J.Z. Bai et al., Phys. Lett. B472 (2000) 200. 20. A. Etkin et al., Phys. Lett. B201 (1988)

Lett. B458 (1999) 511. 24. F.E. Close, G. Farrar and Z.P. Li, Phys. Rev. D55 (1997) 5749. 25. C.J. Morningstar and M. Peardon, D60 (1999) 034509. 26. D.V. Bugg, M. Peardon Lett. B (in press).

REFERENCES 1. 2. 3.

A.V. Anisovich

et al., Phys. Lett. B452 (1999)

180; Nucl. Phys. A651 (1999) 253. A.V. Anisovich et al., Phys. Lett. B471 (1999) 271; Nucl. Phys. A662 (2000) 319. A.V. Anisovich et al., I = 0 C = +l mesons from 1920 to 2410 MeV, Phys. Lett. B (to be

4.

published). A.V. Anisovich

et al., Phys. Lett. B452 (1999)

5.

173. A.V. Anisovich

et al., Phys. Lett. B452 (1999)

6.

187. A.V. Anisovich

et al., Phys. Lett. B476 (2000)

7.

15. A.V.

et al., Data on jip

8.

for m.asses 1960 to %jlO MeV, Phys. Lett. B (60 be published). A. Hasan et al., Nucl. Phys. B378 (1992) 3.

9.

Anisovich

E. Eisenhandler 109.

-+

~‘T’x’)

et al., Nucl. Phys. B98 (1975)

568.

21. J.Z. Bai et al., Phys. Lett. B440 (1998) 217; Phys. Lett. B476 (2000) 25. 22. G.F. Xu, Nucl. Phys. A675 (2000) 337~. 23. D.V. Bugg, L.Y. Dong and B.S. Zou, Phys.

Phys. Rev.

and B.S. Zou, Phys.