Is the threshold enhancement in γγ → ϱ0ϱ0 evidence for an isoscalar, JPC = 0++ exotic state?

Volume 109B, number 6

PHYSICS LETTERS

11 March 1982

IS THE THRESHOLD ENHANCEMENT IN ¥ ¥-~ pOp0 EVIDENCE FOR AN ISOSCALAR, jec = 0++ EXOTIC STATE? Rohini M. GODBOLE and K.V.L. SARMA

Tata Institute of Fundamental Research, Bombay 400005, India Received 13 October 1981

The threshold enhancement in the cross section 3"3'-" pOpO, including the lower energy point reported by the CELLO group, and the observed angular distributions have been interpreted in terms of an isoscalar resonance with jPC = 0++, mass 1450-1550 MeV and width 100-150 MeV.

Experimental studies of the process 3'3' -+ pOpO [ 1 - 3 ] have revealed a strong threshold enhancement with values of the cross section exceeding the estimates of the vector-dominance model [4] by a factor 1 0 - 1 0 0 in the c.m. energy range 1.5--2.0 GeV. It is, however, noteworthy that the type of rapid decrease of cross section as the energy increases away from the broad threshold for 2;) 0 production is suggestive of a resonance mechanism. Obviously such an underlying resonance which couples to 27 and to 200 states must have e v e n - C , / = 0 or 2 , J P= 0 ±, 2 -+, ..., and a mass around 2m o. The first observations of the TASSO group [1 ] have already been studied postulating isoscalar resonances w i t h J PC= 0 -+ [5] and jPC = 2 - + [6]. The recently reported data of the CELLO collaboration [2] reveal a 0~ distribution which is quite consistent with isotropy, and a dependence on the helicity decay polar angle OH which supports the hypothesis of positive parity, indicating an underlying 0+ resonance. In the following we interpret the observed threshold enhancement in the reaction 77 -~ pOpO as due to the excitation of a resonance R with the simplest quantum numbers, I G = 0 + and jPC = 0++. The fact that such a resonant activity is absent in the data on 77 -+ 7r+Tr- [2,7,8] in the same energy region, leads us to speculate that the resonance R is a candidate for a low-lying glueball. The coupling of the 0 ++ resonance R to the 27 state and to the 2/) state will be described by the 504

effective interaction (1)(7) F(2) H = la61 [gTF~v uu (7)

+ go Fu(~)(,°) "Fu({)(P)I R,

(I)

where Fur(7 ) = OvAu - 0uA v is the usual electromagnetic tensor, F(p) is the analogous tensor for the isovector field p, and V0 is a mass parameter which is inserted to make the coupling constants gv and go dimensionless , t . We shall choose ~t0 to be 1 GeV in our numerical work. The cross section for the reaction 3'7 -+ R -+ p 0 ( m l ) p0(m2) proceeding through the scalar resonance R is given by

F(R-~vV, s) r(R-,p°p °, s)

(2) '

where the parameter mR denotes the peak value of the mass and FR the full width at half-maximum of the Breit-Wigner resonance. The two partial widths occuring in the numerator according to eq. (1) are:

['(R

-+

7% s)

=

[(g3"/lao)2/16~] s 312 ,

(3)

F ( R -+ pOpO, s) = [(gphao)2/aTrl

X k3(1 + 3 m ~ m 2 / 2 s k 2 ) ,

(4)

, l A simpler coupling of R to the p-mesons, namely, hp(p~~) • [1(2))R will not be considered as it leads t9 a helicity angular distribution which is maximal at OH = 0 and minimal at OH = ~[2, which is exactly opposite to the observations [ 1,2]. 0 031-9163/82]0000-0000]$02.75 © 1982 North-Itolland

Volume 109B, number 6

PHYSICS LETTERS

hence the cross section of eq. (2) would grow linearly with s. To suppress this bad behaviour in the resonance tail for x/s - mR ~ UR, it is customary to introduce a form factor; following ref. [5], we shall modify eq. (6) by defining

where V's is the mass of the resonance R, the masses m l and m2 are the (variable) masses of the final p 0 resonances, and k is the magnitude of the three-momentum o f the p in the rest frame of the two p's. The finite widths o f the p mesons will be taken into account by convoluting the eq. (2) with independent Breit-Wigner shapes Bo(m2) for each of the p mesons,

Bo(m2)= im 2 _

OR(S) =

(5)

where P p ( m ) i s the m-dependent width o f the p-wave decay P -+ 2rr [9]

(m2-4m213/2 m p = 0.776 GeV,

m2-4m 2 :(m o +m 2) - 4m 2

P0p

Fop = 0.l 58 G e V .

Thus by including the finite widths of the p's we obtain on(s) = N2f × Bp(m 2)

dm21Bp(m21)f dm~ OR(S,m2, m2),

(63

where the limits of integration are chosen to cover one width on either side of the central value of the mass

(rap - P0a) ~< mr, 2 ~< (ma + r o D ,

(7)

and the factor N 2 arises from the unit normalization o f each Bo when integrated over the specified range. Obviously it would be unrealistic to expect the above formulation to hold good as we go to energies which are a few widths beyond the position o f the resonance R. Each o f the partial widths given by (3) and (4) goes like s 3/2 at large enough energies, and

(/3 > / 0 ) .

(8)

BR(R --* n+rr - ) F ( R -+ 3`9") < 1.5 k e V , the limit getting more stringent for higher masses of the resonance. Using our values from table 1 we get

Table 1 Representative values of fitted parameters, along with values or" BR (R ~ defined in eq. (11).

2 1 0

OR(S)(m2/sf

This of course does not affect the fit at the low-energy side, s < m 2, because o f the presence o f strong kinematic suppression for a threshold resonance, nor does it affect the angular distributions which depend on the s p i n - p a r i t y assignments of the resonance. The existing data on a(77 ~ pOpO)have sizeable errors and it is not difficult to obtain a fit for them. We have adjusted the three parameters mR, PR and (gvgp/la20)for a fixed value of the form-factor exponent/3 to obtain the fits. Some representative sets of the parameters are listed in table 1. With ¢3= 0, values o f m R ~< 1.5 GeV will not fit the data because then the increase o f the cross section at higher energies shown in fig. 1 sets in at smaller values ofx/s. A nonzero value of/3 is needed to fix m R < 1.5 GeV (which is suggested by the large cross section in the lowenergy bin according to CELLO group), and we get an overall fit to the data for m R in the range 1 . 4 5 1.55 GeV and width FR within 1 2 0 - 1 5 0 MeV. The values o f the products of the coupling constants have been recast as products of BR(R - , p0p0) F ( R -+ 77) in units of keV. Since BR(R -+ pp) = 3 B R ( R --, pOpO), the values of table 1 imply a lower limit of about 6 keV on F ( R - ~ 73'). The data on 77 + n +Tr- [8] ~ve a limit on the production o f an isoscalar resonance with jPC = 0++ in the mass range 1 3 0 0 - 1 4 5 0 MeV as

mp r' e (m) m2)2 + [m a pp(m)-]g,

l l March 1982

mR

PR

gygp/p~

(MeV)

(MeV)

(GeV) -2

BR(R (keV)

1450 1480 1530

150 120 120

0.078 0.055 0.036

2.1 2.4 2.5

~

pOpO)I"(R -~ 3"1') in keY and energy averaged values of c~

pOpO)F R ~ 9 , ~

a(TASSO)

a(CELLO)

0.51 0.52 0.52

0.53 0.56 0.60 505

Volume 109B, number 6 I

T

PHYSICS LETTERS I

I

I

I

The t o t a l cross section is o b t a i n e d b y i n t e g r a t i n g this a n d c o n v o l u t i n g it w i t h the r e s o n a n c e shapes o f t h e t w o p m e s o n s as in eq. (6). The V D M a m p l i t u d e b e i n g d i f f r a c t i o n a l is pure i m a g i n a r y , in c o n t r a s t to the r e s o n a n t a m p l i t u d e w h i c h is m o s t l y real in t h e tail o f the Breit- Wigner resonance. Thus an a d d i t i o n o f t h e s e t w o a m p l i t u d e s ( a s s u m i n g t h a t VDM provides t h e s m o o t h b a c k g r o u n d u n d e r the r e s o n a n c e ) a m o u n t s a p p r o x i m a t e l y to a n i n c o h e r e n t a d d i t i o n o f cross sections, w i t h very little i n t e r f e r e n c e b e t w e e n t h e m . However, in view o f the small m a g n i t u d e o f t h e V D M c o n t r i b u t i o n s h o w n in fig. 1, such an i n c o h e r e n t a d d i t i o n h a r d l y affects t h e r e s o n a n c e fit to the t o t a l cross section. On the o t h e r

I

I3 CELLODATA i

TASSO

11 March 1982

DATA

I00

~75

(I.

b 5O

I

I

I

a) 25

-

-"-N/~-a

I

I 1.4

I.co

1.8

2.0 2.2 (GeV)

--Ib

2.6

°[

~-

C

I 0.2

0.4

I

I

I

(10)

B = 5.6 ( G e V ) - 2 .

4z We i~ave not included the MARK II data [3] in our fits as their measurements on 73' -~ 2n +2rr- include a sizable background due to non-resonant final states. 506

I

0.8

I

I

TASSO

I

DATA

(9)

Fig. 1 displays some o f o u r fits t o the d a t a ¢2. Dep e n d e n c e o f o u r fits on/3 is displayed b y t h e t w o curves labeled/3 = 0 a n d 13 = 2. T h e d a s h e d - d o t t e d line at the b o t t o m is t h e e s t i m a t e based o n t h e v e c t o r m e s o n d o m i n a n c e m o d e l ( V D M ) . A c c o r d i n g to this m o d e l t h e d i f f e r e n t i a l d i s t r i b u t i o n in t h e s q u a r e d mom e n t u m t r a n s f e r t b e t w e e n 7 a n d p 0 is expressed as a d i f f r a c t i o n t y p e f o r m u l a [1,4]

A = 1.0,

I

0.6 cosO;

2.0

(do/dt)vDM = 10 --31(4k2/s) A eBt ( c m / G e V ) 2 ,

I

I

0.5

Fig. 1. Best fit to the CELLO and TASSO data on energy dependence of ~(73' -~ p°P°). Solid line corresponds to M R = 1.45, F R = 0.15, ¢3= 2, the dashed line corresponds to M R = 1.53, F R = 0.12,/3 = 0 and the dashed-dotted line shows the VDM contribution.

B R ( R -~ n+~-) < 0 . 2 - 0 . 2 5 . B R ( R --* pp)

I

(.5

-

I VDM 2.4

I

CELLO DATA

bO --Ib 0 5

I 0.2

I 0.4

I 0.6 COS0;

I 0.8

Fig. 2. (a) Predicted cos 0 ; distributions along with CELLO data (averaged over 1.3 ~
Volume 109B, number 6

PHYSICS LETTERS

hand, the presence of a background VDM contribution can show up in the production angular distribution 0 o (c.m. angle between the p and the initial photon directions) as a peak near Op ~ 0 standing o]1 a large isotropic distribution o f the J = 0 resonance as in fig. 2. The CELLO data are quite consistent with an isotropic distribution with no hint of a forward spike. The TASSO data do show a peak in the bin cos 0p = 0 . 8 - 1 . 0 , although this point seems to have come down in the later data (without acceptance correction) averaged over 1.4 ~ x/s ~< 1.7 GeV [8]. Our calculations, indicated by the curves in fig. 2, consist of adding the isotropic resonance decay distribution to the peaked distribution of VDM, smearing with the Breit-Wigner shapes for the p's, and averaging over the appropriate energy ranges, namely, x/} = 1 . 3 - 2 . 3 GeV for the CELLO data and V~ = 1 . 5 - 2 . 0 GeV for the TASSO data. As for the distribution of the helicity decay-angle 0~ (the angle between the decay pion and the p direction as seen in the O rest frame), it is governed by the first spherical harmonic function I Y~ 12. The zero spin o f R implies that the two final p's must have the same helicities. I f P 1 , P 1,P0 are the probabilities for the p to be emitted with helicities 1, - 1 , 0 , then since

11 March 1982 l 1.8

1.4

1

I

I

I

i

i~

1

b t) -o

--Ib

i

CELLO

I

DATA--

"",

o.6 \\ 0.2

--

\--

I

t

0. I

rF'b'l

b --Jb

~

I 0.3

I

i

I

I 05 CO'SOH

l

L

gr

I

I 07 •

i

I

I

I , 0 9 .

I

I

1.0

O.6 \

P1 =P-~l = ~(l - P 0 ) , the ~b-integrated normalized angular distribution is given by W(COS 0~]) = 3Pl(1 -- OLCOs20H),

(1 ]1

where

0 ,2

.

--

I O.t

t

I 0.5

i

( O. 5 COS

(£:.

\-

t H

I 0.7

J

I O. 9

8vr

Fig. 3. (a) Predicted distributions for cos OH, along with

CELLO data (averaged over 1.3 ~
1 -

PolP

t = (3&

-

1)lP~

.

For the case o f VDM, since P0 = 0 one obtains the sin20 distribution; also for the case of a pseudoscalar resonance 0 - considered by Layssac and Renard [5] the same sin20 distribution ensues. However, for our case of scalar resonance R the interaction ( l ) implies P1 is less than 1/2 and is given by: PI =l~ [1 + (nt2,n]/sl,:2)]/[l + 3(m2m]/sk2)] .

(12)

The solid curve in fig. 3 is the result o f eqs. (11) and (l 2), after smearing over the p resonance shapes, and energy averaging as naentioned earlier. The predicted curve depends, although weakly, on the form factor exponent/3 due to the energy averaging; the addition of VDM contribution increases the value ofc~ only by

line shows expected distribution for 0 - + and VDM contribution. Solid line shows the expected distribution for 0++ resonance. (b) Same as in (a), with TASSO data (averaged over 1.5 ~<~ ~< 2.0 GeV).

about 5%. We note that the hypotheses of 0 + versus 0 resonances yield for the 5 data points o f TASSO X2 = 6 compared to 12, and for the 10 data points of CELLO the X2 are 20 and 47. Thus the hypothesis of 0 + is preferred over that of 0 - . In conclusion, the experimental observations on the reaction 7~ ~ pOpO below 2 GeV c.m. energy can be interpreted as proceeding dominantly through )a jPC= 0++ resonance R with a mass 1 4 5 0 - 1 5 5 0 MeV and width 1 2 0 - 1 5 0 MeV. A caveat to be mentioned 507

Volume 109B, number 6

PHYSICS LETTERS

is that the first version o f the T A S S O data [1] contain a point at cos 0 o ~ 0.9 which is difficult to reproduce with a J = 0 resonance, The later T A S S O data [8] as well as the statistically superior data of the C E L L O group [2] are in accord with the hypothesis o f 0 + excitation. We have not considered here the effect o f the tail o f the 2 ++ resonance f ( 1 2 7 0 ) as its inclusion will sensitively depend on the assumed f o r m factor suppression. Finally, the fact that no e n h a n c e m e n t is observed in the 3'7 ~ rr+rr- cross section around ~ = 1.5 G e V [2,7,8], indicates that the R is weakly coupled to the 2rr states and perhaps also to the K K state. Thus the R should have very little mixing with an ordinary I = 0 m e m b e r (such as the e) o f a jPC = 0++ n o n e t , and might be an exotic resonance state such as a glueball. Hence the resonance R should be seen to couple to pp, coco, parr, rTr/, 4rr, 69 etc. in hadronic reactions such as g p annihilations. Bag m o d e l calculations o f the glueball s p e c t r u m [10] predict a 0 ++ state at 1.45 GeV, which is consistent with our interpretation. On the o t h e r hand the 0 - + and 0 ++ states are e x p e c t e d to be degenerate in potential models [11]. I f we consider a 0 - + resonance, say R', w i t h an effective interaction H ' =/~61 e , ~ ,

(1)

X (g3" F,~v (3')

(13) r(2

,

(1)

a# (3,) + gp F~v (P)

.

then we obtain for the partial widths

F(R' -* y~,, s) = [(g'y/lao)2/16rr] s 3/2 , P ( R ' -+ p0p0, s) = [(g'o/120)2/27r] k 3 ,

(14)

comparing this with eqs. (3) and (4) we see that the threshold behaviours for R and R' to decay into two p are different ~3. F o r the a b o v e m e n t i o n e d interesting +3 If we try to fit the present data on o(y3"--* pOpO)with R' alone, with mR P= 1.45 GeV, we need 13>/3, which implies considerable mutilation of the original Breit-Wigner formula.

508

11 March 1982

case o f two degenerate scalar resonances of opposite parity, we have m R = m R' = 1.45 GeV, ['R = PR' = , t t 0.1 5 GeV. Assurmng g3' = g3" and go = gp, then a slightly better fit for the cos oH data in fig. 3 is obtained for the hypothesis o f a degenerate pair. We find that the contribution o f R' to 0(73' ~ pOpO) in fig. 1 is small (smaller by about a factor 4 than the contrib u t i o n of R, at V~ = 1.5 GeV), and hence the h y p o t h esis of a degenerate pair can not be excluded even if the d o m i n a n t features o f the data can be explained on the basis o f a 0 ++ resonance.

References [1] TASSO Collab., R. Brandelik et al., Phys. Lett. 97B (1980) 448. [ 2] Measurement of the reactions 2,3'--* ~r+Tr-rr+rr-, 2,2, __. fo .~ 7r+Tr- with CELLO detector at PETRA, contributed paper Intern. Conf. on High energy physics (Lisbon, Portugal, July 1981). [31 D. Burke et al., Phys. Lett. 103B (1981) 153. [4] T. Walsh, J. de Phys. 35 (1974) C2. [5] J. Layssac and F.M. Renard, Montpellier preprint PM/80/11, PM/8115. [6] H. Goldberg and T. Weiler, Phys. Lett. 102B (1981) 63. [7] PLUTO Collab., Ch. Berger et al., Phys. Lett. 94B (1960) 254; TASSO Collab., E. Hilger et al., DESY report 80/75 (1980); BONN-HE-80-5; DESY report 81/026. [8] E. Hilger, Invited talk Intern. Colloq. on Photon-photon interactions (Paris, April 198l), BONN-HE-81-4. [9] J.D. Jackson, Nuovo Cimento 34 (1964) 1644. [101 J. Donoghue, K. Johnson and B. Li, Phys. Lett. 99B (1980) 416; J. Donoghue, Invited talk, XXth Intern. Conf. on High energy physics (Madison, WI, July 1980). [11} H. Suura, Phys. Rev. Lett. 44 (1980) 1319.