isoscalar resonances in the mass region below 1900 MeV: Observation of the lightest scalar glueball

IUCLEAR PHYSIC~

PROCEEDINGS SUPPLEMENTS

Nuclear PhysicsB (Proc. Suppl.) 56A (1997) 270-274

ELSEVIER

N o n e t C l a s s i f i c a t i o n of S c a l a r / I s o s c a l a r R e s o n a n c e s in t h e M a s s R e g i o n b e l o w 1900 M e V : O b s e r v a t i o n o f t h e L i g h t e s t S c a l a r G l u e b a l l A.V. Anisovich I , V.V. Anisovich 1, Yu.D. Prokoshkin2 and A.V. Sarantsev 1 (1) Petersburg Nuclear Physics Institute, Gatchina, St.Petersburg 188350, Russia (2) Institut for High Energy Physics, Protvino, Serpukhov, 142284, Russia

Here we summarize the results of the investigation of the I J PC = 00 ++ wave [1-3]: this investigation is devoted to the search for the lightest scalar glueball. In Refs. [1, 2], the qq-nonet classification of the scalar/isoscalar states is performed in the mass region below 1900 MeV basing on the following data: GAMS data for rt-p --~ rt°r°n [4], r/~n [5], r/~'n [6]; CERN-Miinich data for r - p --+ ~r+rr-n [7]; Crystal Barrel data for p/~ --+ rr%r%r°, ~r%r°rh r~°~r/ [8, 9]; BNL data for r~rr --+ K K [10]. The nonet classification of the states and search for extra states look like the only way for the identification of lightest glueballs. The 00++-wave is analyzed in ReD. [1, 2] in terms of the K-matrix elements which are presented in the form: (4) (~)

Kij = ~

gi gj m--~: s + smooth t e r m s ,

(1)

(R

where the pole terms (s is invariant energy squared) describe the input meson states "before" the decay and their mixture due to the transition into lrrt, K K , ~ , ~ ' and rr~rlrr; the K-matrix poles are referred as poles of "bare states". The couplings g(~) are related to the input-state couplings of the dispersion relation N/D-amplitude: the Kmatrix couplings obey the same relations as couplings of input poles of the N/D-amplitude [3], thus allowing to analyse the q~]/glueball content of bare states. Coupling constants are determined by leading terms in the 1/No-expansion for the process of Fig. 1: gluons produce new q{-pairs with the flavour symmetry violation, u~ : dd : sg = 1 : 1 : A, with strange-quark production probability suppression factor A ~_ 0.5 [2]. The q~-state with non-strange/strange quark content q~= nhcos¢+sgsin¢, where n~ = (ufz+dd)/v/2,

~ mesonI ~

q~-meso%

meson

G

"~'~

meson

a)

b)

~

meson

meson

gluon"~

meson

c)

Figure 1. Diagrams for the decay o f a q@-meson (a) and a glueball (b,c) into two q@-meson states. decays into two pseudoscalar mesons (Fig.la) with the following normalized production probabilities

[1, 2]2

2

2

% r : 7~:R " 7.~ : % , ' = =

~cos ¢ : ~

~v/-2sin¢+v'~cos¢

:

( os 0co + in Osin )

:

sin2Ocos~O(cos¢-x/~sin¢) 2 .

:

(2)

Here 7~ = Fi/f~i, where Fi is partial width and f~i is corresponding phase space factor. The angle /9 determines the quark content of ,1 and rf. The glueball decay into two pseudoscalar mesons, being a two-meson production process (Fig.lb), has the same couplings as given in Eq.(2) but with fixed angle ¢ - + ¢glueball:

0920-5632/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. Pll: S0920-5632(97)00286-7

~taneglueball = V 2 "

(3)

A. l( Anisovich et al./Nuclear Physics B (Proc. Suppl.) 56,4 (1997) 270-274

25O

200

i

A2('~s-'m~ ")

:~ 150

o 04

~100 7

50

O.5

1

1.5 6) M,,, (OeV)

0.3

~!#

A2(/~'rtKK)

0.2 0.1 0

I.2 I.4 I.6 I.8 b) MKK(GeV)

0.06 0.04

!v

0.02 0

1.2

1.4

1.6 1.8 c) M,~,(GeV)

With Eqs. (2) and (3) for the couplings, a simultaneous fit of the two-meson spectra of Refs. [410] was performed [1, 2] in the framework of the K-matrix representation of the amplitudes. Two solutions describe well the data set: S o l u t i o n I. Two bare states fbare (720± 100) and f b ~ (1260 ± 30) are members of the 13P0 q~-nonet, with f0bare(720) being s~-rieh state: ¢(720) = ¢ba~tl600 ± 50) and - 6 9 ° ± 12 °. The bare states J0 fba~(1810 ± 30) are members of the 23P0-nonet; f b ~ ( 1 6 0 0 ) is dominantly nil-state: ¢(1600) ---- 6 ° + 15 °. The state fboa~'e(1235 ± 50) is superfluous for q~-classification, being a candidate for the lightest gluebalh its couplings to the two-meson decay obey Eq. (3). S o l u t i o n II. Basic nonet members are the same as in solution I. The members of the 23p0-nonet are the following: f0b~re(1235) and f0b~e(1810); both these states have significant sg-component: ¢(1235) = 42 ° ± 10 ° and ¢(1810) = - 5 3 ° 4- 10 °. The state f0bare(1560 ± 30) is supersluous for the q~classification, being a good candidate for the lightest glueball. The description of meson spectra taken from Refs. [4-10] using the solutions I and II is shown in Figs. 2-5. In the region 1000-1900 MeV there are five scalar resonances which are related to the five poles in the complex-mass plane: Resonance

Position of poles on the complex-M plane, in MeV units

1015 ± 15 - i(43 ± 8) ]0(980) 1300 ± 20 - i(120 i 20) f0(1300) 1499 ± 8 - i(65 ± 10) /0(1500) 1780 ± 30 - i(125 ± 70) f0(1750) 15302 °0 - i( 60 ± 140). fo(1200- 1600)

0.04

0.02 0

271

, , I, 1.6

,,

I,,, 1.7

1"~'~ 1.8 1.9

(4)

Comparison of the positions of bare states and those 1.5 of real resonances shows that the observable states d) M,~,/(OeV) are strong mixtures of bare states and two/fourmeson states (into which these resonances decay). Figure 2. Description of the S-wave amplitude For further analysis of the mixing, we have rewritsquared in the reactions: ~r~r --+ ~rlr (data are ten the 00++-wave amplitude in terms of the q~taken from ref. [4]) (a), 7rTr --+ K / f [10] (b), states and input bare states [3]. Although this lrlr -+ ~/ [5] (c) and 7r~r --~ q~' [6] (d). Solid transition from the K-matrix to the amplitude, curves correspond to solution II and dashed which is analytic in the whole s-plane, faces amcurves to solution I. biguities related to the left-hand side singularities

A. E Anisovich et al../Nuclear Physics B (Proc. Suppl.) 56.4 (1997) 270-274

272

of vertices bare state --+ qq, g~(s), it allows us to restore the mass of the pure glueball, that is an object of lattice calculations. In addition, it clarifies the situation with the mixture of overlapping resonances. The point is that in the case of two overlapping and mixing resonances, one resonance accumulates the widths of both initial states (Fbroa~ ~-- F1 + F2), while the other resonance be-

~ o

2O

4oooIA.< l°>b

10 o O4

~

I Ill

0

Ill

0.8

IJilll

1

i

1.2

CN

7

0

1.4 10000

10 -= 0 . 4 5 < - t < 1 . 0

1.5

2

0

2.5 'i'r

I I I I,

I I i,,,

1

1.2

0.8

-

-

-

"~J-

-

-

- ~b

~-,-,-,- ]~t',7 -i-,-I- T~ ; %0.5 1 1.5 2 N -

-

.

.

.

.

OOO

5

0

0.5

1

1.5

2

i

1.4

300

2500

o

"7

>" 200 0

1

/ .

>~ 7.5 o

0.5

5

z

% "-

200OOIi 0

0 . 5 5 < - t < 1.0

7>~1 5

comes narrow (F,~r~o~o --4 0). In the case of three overlapping resonances, two final states have small widths (Fn~rrow-1 --4 0, Fn~rrow-2 --+ 0),

I 0.5

100

1

1.5

2

20004000 I N < Y 4 ° ~ ,

z

0

0.8

0.9

1

1.1 1.2 M,,~ (OeV)

Figure 3. Event numbers versus invariant mass of the ~rTr-system for different t-intervals in the 7r-p --4 7r°rr°n reaction [4]. Solid curves correspond to solution I I and dashed curves to solution I.

0

i I I I,

0.5

1

1.5 2 M,,,(GeV)

Figure 4. Fit of the 7rrr angular-moment distributions in the final state of the reaction rr-p --+ nrr+rr - at 17.2 G e V / c [7]. The curve corresponds to solution II.

273

A. V. Anisovich et al./Nuclear Physics B (Proc. Suppl.) 56A (1997) 270-274

while the third resonance accumulates the widths of all initial states (Fbroad -- F1 -F F2 -Jr F2). The glueball propagator, which takes into account the transitions of Fig.6 type, is determined

by the q~ loop diagrams with the vertices g~(s): 1

= -I x z-I_ _ 0

60

0

r-., 4 0 O4 0

©

20

z

0

,v-4.,'>

s

(5)

,Je~ [(m~ - s)aob - sob(s) I = 0,

'l''''l',,,I,,,kl,,,,IL,

0.5 0.75

% '7 %

-

s'

where # = m:+k~x(1-~) and m is the quark mass. Two types of quark loop diagrams are taken into account: with non-strange (nil) and strange (s~) quarks, their relative weights are given by mixing angles ¢. Complex masses of the physical states are determined as zeros of the determinant:

80 %

(2n)3

1 1.25 1.5 a) M,,, (OeV)

(6)

where ma are the masses of the input meson states and ~i~b is unit matrix.

q O

8

~

a

l

l

q

6

o)

0

Ob

4

(5

2

z

0

o,,I o

G

o.4

0.6

0.8

1.2

b)

M,.., (OeV)

~

a

l

l

Figure 6. Diagrams for mixing glueball/qq mesons.

Within a simple ansatz about the power decrease of vertices 9a(S) at large s, we refit the 00 ++ am% plitude in the mass region 1200 -1600 MeV. Results -k'- 10 for solutions I and II are shown in Fig. 7. To illus% trate the dynamics of mixing, the following method (1) is used: we change 9a --+ ~ga and vary ~ from 0 (no (_9 mixing) to 1 (the real state corresponding to the 5 cO description of data). The pole movement with ~ is o shown in Fig. 7. For both solutions the broad resc~ onance f0(1530+9~0) is the descendant of the pure z glueball. No wonder: the glueball mixes with both 0 1.2 1.4 1.6 q4-states without suppression, while the mixture of the qq-states is suppressed, for they are members of c) M,, 7 ( O e V ) different q4-nonets. Final position of poles after the mixture of initial (or bare) states supports the idea Figure 5. The n%r ° spectra: (a) in the iv# --+ about the mixture of a glueball with neighbouring 7r°Tr°Tr° reaction, (b) in the pp ~ r/n°n ° reac- qq-mesons. The glueball, being a particle of another tion; (c) r/r/spectra in the p/~ --+ n°r/r/reaction. origin than 3 P 0 q~-states, sets foot in the series of Curves correspond to solution II. q~ states and mixes with neightbouring mesons: an

A. E Anisouich et al./Nuclear Physics B (Proc. Suppl.) 56A (1997) 270-274

274

existence of a broad state is an inevitable result of such an "invasion". The broad resonance (the descendant of the pure glueball) has the following q4-meson/glueball content in solutions I and II correspondingly: f0(1530+9°0) -+ 28%(q4)1 + 27%(q4)2 + 45% G; f0(1530+9°0) --+ 10%(q~)1 + 42%(q~)2 + 48% G. The mass of the pure glueball in the solution II (~ = 0) is equal to: mp,,~e gl,,eb~u

=

(7)

1695MeV

This value agrees with the results of lattice gluodynamics: 1600 + 85 + 100 MeV [11], 1740 =h 70 MeV [12]. Present analysis allows to conclude: in the region 1300-1600 MeV we have the scalar glueball. The pure glueball state has mainly dispersed over three real resonances: f0(1300), f0(1500) and fo(1530+9°o).

~,

0

~-0.1

E ~=o.5

I -0.2 -0.5

References [1] V.V. Anisovich and A.V. Sarantsev, "K-matrix analysis of the I J P c = 00 ++ amplitude in the mass region up to 1550 MeV", Phys. Lett. B., in press. [2] V.V. Anisovich, Yu.D. Prokoshkin and A.V. Sarantsev, "Nonet classification of scalar/isoscalar resonances below 1900 MeV: the existence of an extra scalar state in the region 1200-1600 MeV", Phys. Lett. B., in press. [3] A.V. Anisovich, V.V. Anisovich and A.V. Sarantsev, "The lightest glueball: Investigation of the 00 ++ wave in the dispersion relation approach", to be published. [4] D. Alde et al., Z. Phys.C66 (1995) 375; A.A. Kondashov et al., Proc. 27th Intern. Conf. on High Energy Physics, Glasgow (1994) 1407; Yu.D. Prokoshkin et al., Physics-Doklady 342 (1995), 473; A.A. Kondashov et al, Preprint IHEP 95-137, Protvino (1995). [5] F. Binon et al., Nuovo Cim. A78 (1983) 313.

-0.4 Solution I

:

[6] F. Binon et al., Nuovo Cim. A80 (1984) 363.

3 , ~ ~:=1.0

-0.5 1.1

1.2

1.3

1.4

1.5

1.6

~

0 ~-0.I

1.8

~0.

[7] B. Hyams et al., Nucl. Phys. B 6 4 (1973) 134. [8] V.V. Anisovich et al., Phys. Lett. B 3 2 3 (1994) 233. [9] C. Amsler et al., Phys. Lett. B 3 4 2 (1995) 433. [10] S.J. Lindenbanm and R.S. Longacre, Phys. Lett. B 2 7 4 (1992) 492; A. Etkin et al., Phys. Rev. D 2 5 (1982) 1786.

-0.5 I -0.4 -0.5 i

1

1,7

Solution II (1110/` ,,l,,,l,,,I,,,J,,,l,,~l,,,l,,i 1.1

1.2

1..~

1.4

1.5

1.6

1.7 1.8 M (CeV)

Figure 7. Trajectories of poles in the complex M-plane with increase of coupling constants: g~ --4 ~g~.

[11] F.E.Close and M.J.Teper "On the lightest scatar glueball", RAL-96-040 (1996). [12] J. Sexton, A. Vassarino and D. Weingarten, Phys.Rev.Lett. 75 (1995) 4563.