qq -state mixing in the mass region near 1500 MeV

6 March 1997

PHYSICS LElTERS B Physics Letters B 395 ( 1997) 123- 127

ELSEWIER

Of

+-glueball/@-state

A.V. Anisovich,

mixing in the mass region near 1500 MeV V.V. Anisovich,

A.V. Sarantsev

St.Petersburg Nuclear Physics Institute. Gatchina, St. Petersburg 188350, Russia

Received 5 November 1996; revised manuscript received 10 December 1996 Editor: L. Montanet

Abstract Based on the results of the K-matrix fit of the (IJ pc = OO++) wave [V.V. Anisovich, Yu.D. Prokoshkin and A.V. Sarantsev, Phys. Len. B 389 (1996) 388; V.V. Anisovich and A.V. Sarantsev, Phys. Lett. B 382 ( 1996) 4291, we analyze

the analytic structure of the amplitude and qqlglueball content of resonances in the mass region 1200-1900 MeV, where an extra state for q+systematics exists being a good candidate for the lightest scalar glueball. Our analysis shows that the pure glueball state is dispersed over three resonances: fo( 1300), fo( 1500) and fo( 1530Twm), while the glueball admixture in fo( 1750) is small. The broad resonance fo( 1530?ym) is the descendant of the lightest pure glueball. The mass of the pure glueball is 1630 f z MeV, in agreement with Lattice calculation results [F.E. Close and M.J. Teper, On the lightest scalar glueball, preprint RAL-96-040 (1996); G.S. Bali et al., Phys. Lett. B 309 (1993) 378; J. Sexton, A. Vaccarino and D. Weingarten, Phys. Rev. Lett. 75 (1995) 45631.

1. Introduction In Ref. [ 1 ] the K-matrix analysis of OO++-wave has been performed in the mass region 500-1900 MeV for the channels i = ITIT, Ki?, qq,q$ and TUCTIT. The matrix elements Kijin the fit of Ref. [ I] were presented as a sum of poles, c, gja)gl”)/( M: - s) , and smooth background terms, fij: the k-matrix poles correspond to input states. Partial amplitude poles which correspond to observable states, are determined by the mixture of input states via thier transition into real mesons. Therefore, the wave function of a physical state is a mixture of not only the input states but also of real mesons. Because of that input states are called “bare states” in Ref. [ 11, i.e. the states without a cloud of real mesons. Decay coupling constants of bare states are fixed by their quark-gluon content [4-61. Simul-

taneous fit of the data of GAMS [ 7,8], Crystal Barrel Collaboration [ 91, CERN-Miinich collaboration [ lo] and BNL group [ 111 fixes five K-matrix poles (or five bare states). Only two of them are definitely &rich states: fta;“( 720 f 100) and fE”( 1830 f 30). For other three states, fk=‘“(1230f50), fi=( 1260f30) and fi”( 1600 f 50) : two of them are natural qQ nonet partners for f$“( 720) and fi”( 1830)) while one state is extra for qq-systematics. The analysis of Ref. [ l] gives two solutions which describe the data set well: Solution I: fiUe(720) and fk=(1260) are 13Po nonet partners, fi”( 1600) and fi’“( 1810) are 23Po nonet partners, $“( 1230) is a glueball;

0370-2693/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. P/I SO370-2693(97)00006-3

124

A.V. Anisovich

et d/Physics

Letters B 395 (1997) 123-127

Solution II: foXe( 720) and fibare( 1260) are I 3Po nonet partners, fobare(1230) and fgXe( 18 10) are 23Pa nonet partners, fk”( 1600) is a glueball. Physical states are mixtures of bare states which occur in the K-matrix formalism via transitions of bare states into meson channels (in the analysis of Ref. [ 1] : zvr, KI?, 7,177, r/v’ and 47r). In the mass region 12001650 MeV which is key region for determination of the lightest glueball, the OO++-amplitude has four poles at the following complex masses (in MeV) [ 1 ] :

(1300 f 20) (1499*8) (1530~~~,) (1780 f 30)

-

i( 120 f 20)

--+ + i(560f 140) + i( 125 & 70) -+

i(65+10)

b)

Cl

d)

fe( 1300) fa(1500) fa( 1530+;!,) fc( 1780) Fig.

(1)

I. Diagrams which provide the glueballly~

mixing.

1

Each of these states is a mixture of q4 and glueball components. In order to reconstruct qqlglueball content of the resonances, we have performed here a reanalysis of the OO++-amplitude in the region 12001650 MeV using the language of q4 and glueball states.

1

&b(S)

= @--+

0

Mixing of q@states with glueball is due to the processes shown in Figs. la, lb: gluons of the glueball produce a qq-pair, Fig. la; the produced quarks interact by gluon exchanges, Fig. lb. According to the rules of the 1/N-expansion [ 121, the main contribution into the interaction block is given by planar diagrams. Saturating the qij-scattering block of Fig. lb by q4 states, we represent the diagrams of Fig. lb as a set of diagrams of the type shown in Figs. Id and le. The sum of diagrams of Fig. lc, Id, le, and so on gives the glueball propagator with q+state mixing taken into account. The mixtures of the pure glueball state and input qij-states are determined by the quark loop transition diagrams, Bab( s), (s = p2 is glueball four-momentum squared) which have the following form in the light cone variables:

d2kL &l(s’)gl7(s’)

s

x 2(s’-4m2).

s’ - s - i0

(2)

Here s’ = $,T’, g, and gb are the vertices of the transitions state a + q4 and state b -+ q4, and m is quark mass. The factor 2( s’ - 4m2) is determined by the spin structure of the quark loop diagram: Tr[ (R + m)(-f?+I+m)]

2. Glueball propagator

.I

dx x

=2(s’-4m2).

To analyse analytic structure of the OO+‘-amplitude in the mass region of the resonances of Eq. ( 1), let us introduce a 4 x 4 propagator matrix, Dab(s), which describes the transition state a --t state b with a, b = 1,2,3,4, in accordance with the number of investigated states. The diagrams of Fig. 1 type give: D;b’(s)

= (mi -

s)&,b --&b(S).

(3)

Here m, is an input mass for the state a; in the case of the glueball it is the mass of pure gluonic glueball, C&bis unit matrix, and Bab( s) is given by Eq. (2). The zeros of the determinant II(s)

= detl(mi

- s)&,b -&b(S)/

(4)

determine the complex masses of physical states: in the case under investigation, they are given by Eq. ( 1) . Let us denote them as MA, MB MC and MD. Then,

125

A.V. Anisovich et al./Physics Letters B 395 (1997) 123-127

in the vicinity of s = Mi, Dob( s) is described by the pole term only:

Dab(SN Mi)

NA

N

E,

(5)

A

where four coefficients straint

(21, a~, ~3, a4 satisfy the con-

and determine the probabilities of the input states ( 1,2,3,4) in the physical state A; NA is a normalization factor common for all Dab. In order to take into account the flavor content in the quark loop diagrams which is omitted in Eq. (2)) the following replacement should be done: Bab( s) --) cos &J cos &,B$@ ( s)

,

•+ sin &, sin &&2)(s)

(7)

where ]a) = cos & nii + sin &, SS and lb) = cos C$bnS + sin C,&sS, while BLy’ and B68’ refer to loop diagrams with non-strange and strange quarks, nA = (uii + dd) / fi and sS. For pure glueball state, a = glueball, the effective mixing angles are determined by relative probabilities of the production of non-strange and strange quarks by gluons, ufi : dd : sS=l: 1: A, so that tg &iuet,alt = m. Experimental data give A N 0.5 [ 6,131: this value corresponds to &ue~ait N 25”. Mixing angles for 13Paq4 and 23Poqij states were found in Ref. [ 11.

3. Fit of the OO++ amplitude For the calculation of loop diagrams, Eq. (2), we should fix the vertices g,(s) . We parametrize the vertices for the transition state a + nii in a simple form:

:

13Pa q+state

g](s) =y1 fi

Glueball

=y2

:

-

k2 + a2

83(S)

23Pe q&second

_d

k$t-a2

$5

=73

state :

*

1

k; +u2 k2 + cr2 + h ’

k; + ~3 -* k2 + c3 ’

g4( s) = g2 (s) .

W,(A) = Iba12.

(9)

The probabilities W, are given in Table 1 together with masses of physical resonances, MA, and masses of input states, m,.

4. GluehalUq@state

mixing

In order to analyze the dynamics of the glueball/qq mixing, we use the following method: in the final formulae the vertices are replaced in a way: &Z(s) + 5&(s)*

(10)

with a factor 6 running in the interval 0 5 5 5 1. The case 6 = 0 corresponds to switching off mixing of the input states. The input states are stable in this case, and corresponding poles of the amplitude are at sa = rn$ Fig. 2 shows the pole position at 5 = 0 for solution I (Fig. 2a) and solution II (Fig. 2b). For glueball state mg is the mass of a pure glueball, without qq degrees of freedom. In solution I the pure-glueball mass is equal to

k; + CI -. k2 + uI ’

23Pe q&first state : g2(s)

Here k2 = $ - m2 and k2a = d4 - m2* ? ma, ya and u. are parameters. Masses of constituent quarks are equal to: m = 350 MeV and m, = 480 MeV. Factor d is due to orthogonality of the 13Peq4 and 23Poq4 states: we put Re&( so) = 0 at & = 1.5 GeV. (In the case of s-dependent B-functions the orthogonality requirement for loop transition diagrams cannot be fixed at all values of s) . The parameters m,, a,, h and Ya (a = 1,2,3) are to be determined by masses and widths of the physical resonances of Eq. ( 1). However, the masses m, are approximately fixed by the K-matrix fit of Ref. [ 11, where masses of the K-matrix poles, Mime, are determined: (A4i”)2 -m~-B~,((M~~)2).Letusstress that m3 is the mass of pure gluonic glueball which is a subject of Lattice QCD calculation. Parameters which are found in our fit of the OO++ amplitude in the mass region 1200-1900 MeV are given in Table 1. Using these parameters, we calculate the couplings (Y, which are introduced by Eqs. (5) and (6) : these couplings determine relative weight of the initial state a in the physical resonance A:

(8)

mpure gluebarl(Solution I) = 1225 MeV,

(11)

126

A.V. Anisovich

Table I Masses of the initial states, coupling

Solution I h = 0.25 GeV2 d= 1.01

and @/glueball

Letters B 395 (1997) 123-127

content of physical

initial state

states

13P” n&rich IpI = Iso

239 n&rich

!?Ii (GeV)

1.457

yi ( GeV3f4 ) vi (GeV*)

0.501 0.075

Glueball 43 = 25’

23Po s&rich 44 = 84’

1.536

1.230

1.I50

I.040 0.225

0.320 0.375

1.040 0.225

32% 25% 44% 1%

12% 70% 24% 1%

55% 3% 27%

1% 2% 4% 98%

Initial state

l”4, n&rich 4I = 18O

23Pu nii-rich 42 = 35O

Glueball 43 = 25’

23P” &rich c#J4= -55O

,ni (GeV) yi ( GeV314) ai (GeV2)

1.107 0.362 0.175

1.566 0.702 0.275

1.633 0.315 0.375

1.702 0.702 0.275

35% 1% 12% 0.1%

26% 64% 41% 0.2%

38% 35% 47% 0.2%

0.4% 0.4% 0.3% 99.5%

W[f0(1300)] W[fo( 1500)] W[fo( 1530)] W[fo( 1780)] Solution 11 h = 0.625 GeV2 d= 1.16

constants

et al./Physics

W[fo( 1300)] W[fu( 1500)] W[fu(1530)] W[f0(1780)]

1.300-iO.115 1.500 - iO.065 1.450 - iO.450 1.780 - iO.085

1.300 - iO.115 1.500 - iO.065 1.450-iO.450 1.750-iO.100

(GeV) (GeV) (GeV) (GeV)

(GeV) (GeV) (GeV) (GeV)

that definitely disagrees with the Lattice-Gluodynamits calculations for the lightest glueball. In solution II mpureglU&,ll(Solution II) = 1633 MeV.

(12)

This value is in a good agreement with recent LatticeGluodynamics results: 1570 f 85( stat) f lOO( syst) MeV [ 21 and 1707 f 64 MeV [3]. With increasing .$ the poles are shifted into lower part of the complex mass plane. Let us discuss in detail the solution II which is consistent with Lattice result. At 5 2 0.1-0.5 the glueball state of solution II is mixing mainly with 23Pa qq-state, at 5 N 0.8-1.0 the mixture with 13Pu qq-state becomes significant. As a result, the descendant of the pure glueball state has the mass M = 1450-i450 MeV. Its gluonic content is 47% (see Table 1) . We should emphasize: the definition of W, suggests that zA=1,2,3,4 Wslueball(A) + 1 because of the s-dependent B,b in the propagator matrix. Hypothesis that the lightest scalar glueball is strongly mixed with neighbouring q4 states was discussed previously (see Refs. [ 14,151, and references therein). However, the attempts to reproduce a quantitative picture of the glueball/qq-state mixing and the mass shifts caused by this mixing could not be suc-

$q = -6’

cessful within standard quantum mechanics approach that misses two phenomena: (i) Glueballlqq-state mixing described by propagator matrix can give both a repulsion of the mixed levels, as in the standard quantum mechanics, and an attraction of them. The latter effect may happen because the loop diagrams Bat, are complex magnitudes, and the imaginary parts Im B,b are rather large in the region 1500 MeV. (ii) Overlapping resonances yield a repulsion of the amplitude pole positions along imaginary-s axis. In the case of full overlap of two resonances the width of one state tends to zero, while the width of the second state tends to the sum of the widths of initial states, Ifirst = 0 and rsecond = rl + r2. For three overlapping resonances the widths of two states tend to zero, while the width of the third state accumulates the widths of all initial resonances, Ibird 2 I1 f I2 + I3. Therefore, in the case of nearly overlapping resonances, what occurs in the region near 1500 MeV, it is inevitable to have one resonance with a large width. It is also natural that it is the glueball descendant with large width: the glueball mixes with the neighbouring 13Pe qq and 23Pc q4 states, which are both nii rich,

A.V. Anisovich et al./Physics Letters B 395 (1997) 123-127

5J

127

6. Conclusion

0

3

"-0.1

The lightest gluodynamic glueball is dispersed over neighbouring resonances mixing mainly to 13Pe qq and 23Pa qq states. With this mixing the glueball descendant transforms into broad resonance, fo( 1530”;;“). Thi s resonance contains (40-50) % of the glueball component. Another part of the glueball component is shared between comparatively narrow resonances, fa( 1300) and fe( 1500) which are descendants of 13Po qc_jand 23Po q4 states.

Y ? -0.2 -0.3 -0.4 -0.5

0

<; s "-0.1 !",

Acknowledgements

y-0.2

We thank T. Barnes, D.V. Bugg, F.E. Close, L.G. Dakhno, L. Montanet and Yu.D. Prokoshkin for useful discussions. This work was supported by RFFI grant N96-02- 17934 and INTAS-RFBR grant N95-0267.

-0.3 -0.4 -0.5 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

References

M(GeV) Fig.2. Complex-& plane (M = Re&, -I’/2 = lm& : location of OO++amplitude poles after replacing gn + &gR.The case [ = 0 gives the positions of masses of input 9q states and gluodynamic glueball; 5 = 1 corresponds to the real case.

without suppression.

5. Problems

Despite the fact that the lightest scalar glueball is now understandable in its principle points, there are problems which need to be clarified. First, it is nccessary to estimate the influence of the more distant resonance, fa (980). Second, the analysis should be repeated in terms of hadron states, without using the language of qq-states. Such an analysis would give the possibility to check the idea of quark-hadron duality in the mass region 1000-2000 MeV which is used here. There is one more problem: K-matrix analysis of OOff wave [ 11 provides two solutions. Correspondingly, analyzing here two variants we rejected one solution basing on the results of Lattice-Gluodynamics calculations. The problem is if it is possible to discriminate between solution I and solution II and what type of experimental data are needed for that.

111 V.V. Anisovich, Yu.D. Prokoshkin and A.V. Sarantsev, Phys.

l&t. B 389 (1996) 388; V.V. Anisovich and A.V. Sarantsev, B 382 (1996) 429. 121EE. Close and M.J. Teper, On the lightest scalar glueball, preprint RAL-96-040 (1996); G.S. Bali et al., Phys. Lett. B 309 (1993) 378. 131 J. Sexton, A. Vaccarino and D. Weingtuten, Phys. Rev. Lea. 75 (1995) 4563. 141 S.S. Gershtein, A.K. Likhoded and Yu.D. Prokoshkin, Z. Phys. C 24 (1984) 305. 151 C. Amsler and EE. Close, Phys. Lett. B 353 (1995) 385. [61 V.V. Anisovich, Phys. l&t. B 364 (1995) 195. r71 D. Alde et al., Z. Phys. C 66 (1995) 375; A.A. Kondashov, Yu.D. Prokoshkin and S.A. Sadovsky, Preprint IHEP 95-I; Yu.D. Prokoshkin et al., Physics-Doklady 342 (1995) 473. [81 E Binon et al., Nuovo Cim. A78 (1983) 313; A80 (1984) 363. 191 V.V. Anisovich et al., Phys. Lett. B 323 (1994) 233; C. Amsler et al.. Phys. L.&t. B 342 (1995) 433. [lo] B. Hyams et al., Nucl. Phys. B 64 (1973) 134. [ 111 S.J. Lindenbaum and R.S. Longacre, Phys. Lett. B 274 ( 1992) 492; A. Etkin et al., Phys. Rev. D 25 (1982) 1786. [ 12) G. ‘t Hoot?, Nucl. Phys. B 72 (1994) 461; G. Veneziano, Nucl. Phys. B 117 (1976) 519. 131 V.V. Anisovich, M.G. Huber, M.N. Kobrinsky and BCh. Metsch, Phys. Rev. D 42 ( 1990) 3045. 141 C. Amsler and EE. Close, Phys. Rev. D 53 (1996) 295. 151 V.V. Anisovich, Physics-Uspekhi 38 (1995) 1202.