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Coupled channel analysis of JPC = 0++and 2++ isoscalar mesons with masses below 2.0 GeV
Physics Letters B 274 (1992) 492-497 North-Holland
P H YSIC S 1 FT T ERS B
C o u p l e d c h a n n e l analysis o f j e c = with m a s s e s b e l o w 2.0 G e V S.J. Lindenbaum a,b and R.S. Longacre
0 + + and 2 + + isoscalar m e s o n s
a
a Brookhaven NationalLaboratory, Upton, N Y 11973, USA b City College of New York, New York, N Y 10031, USA
Received 14 July 1991
A coupled channel analysis of the j e c = 0 ++ and 2 ++ channels below 2 GeV has been performed using Breit-Wigners considering nn scattering amplitudes, J/V radiative decay, and KI~-,KK. The main conclusions were that a good fit with a minimum number of meson poles (four) was obtained in the 0 + ÷ channel. In our fit the fo(1720) was used to describe the first discovered S*' (1720) and the 0 (1720) both of which have the same quantum numbers and parameters to a high probability. Unitarity effects which require a long-termeffort were not included, however they are expected to have important effects primarily near thresholds and thus not affect the major conclusionsof this paper.
1. Introduction
A coupled channel analysis is much more constrained and thus more likely to reveal the m i n i m u m n u m b e r of real resonances required to explain the considerable data available. Therefore, a coupled channel analysis has been performed using BreitWigners considering the n n scattering amplitudes, J / V radiative decay and KI~-,KI~. The spectrum of isoscalar mesons with J P C = O + + and 2 ++ are extracted from pseudoscalar-pseudoscalar final states. It is found that three meson poles are necessary in order to fit the 2 ++ data (1"2(1270), f ' ( 1 5 2 5 ) , and f2 ( 1810) ), while four meson poles are necessary in order to fit the 0 ++ data (fo(975), fo(1300), fo(1400), and f0(1720) ) up to 2.0 GeV. This analysis uses the fact that the f2(1720) or 0 ( 1 7 2 0 ) that is observed in the J/V radiative decay has spin 0, not spin 2. This was found by the M K III Collaboration to be highly probable [ 1 ]. Earlier fits [2 ] (which are still applicable) with the 0 having spin 2 did not agree with the LASS data [3]. A fit was achieved without the fo(1590) while the phase shifts This research was supported by the US Department of Energy under Contract Nos. DE-AC02-76CH00016 and DE-AC0283ER40107. 492
which lead to the claim for this resonance are well described using the fo(1300), fo(1400), and the fo(1720). The 1"o(1300) has a large width and does not show sufficient phase motion, thus it could possibly be due to an unitary effect at the rlq threshold. In our fit the fo(1590) is made up of a sum of the fo(1400) and the fo(1720).
2. Coupled channel analysis of j P C = 0 + + and 2 ÷ ÷
When a pion strikes a proton thereby changing it into a neutron, the low m o m e n t u m transfer It' I scattering is d o m i n a t e d by one pion exchange (OPE). Using the standard OPE extrapolation [4], one obtains the S-matrix Argand amplitudes for each of the partial waves. The spectrum of i soscalar mesons in the j p c = 0 + + and 2 + + channels is very important in testing q u a r k gluon and g l u o n - g l u o n confined Q C D theory. Both qdl and gg states can occur in these channels. However, they can mix, and therefore only a detailed analysis of these mesons in different production and decay channels can determine their nature.
0370-2693/92/$ 05.00 © 1992 Elsevier SciencePublishers B.V. All rights reserved.
Volume 274, number 3,4
PHYSICS LETTERS B
For the following S-matrix analysis we have used the processes nn--,nn, nn--,KK, nn--,qq [2,5] ~, nn--,~n' [6]. We also added the reaction KI~--,KI~ [3] by assuming that the unnatural exchange for M = 0 amplitude for K - p - + ~ ° [3] is pure K exchange. The channels J / t ~ T n + n -, J/~--,TK+K -, J/~-q,K~I~, J/~c--,Tqrl, and J/w-,yqrl' [7] were also used in the S-matrix analysis because they only contain isoscalar mesons with jpc= 0 + + and 2 + + below the mass of 2.0 GeV. We proceed to obtain the S-matrix, which describes meson-meson scattering for the quantum numbers I a = 0 + , JPC=O++ and 2 ++. In particular we concentrate on the pseudoscalar-pseudoscalar couplings. The T-matrix is related to the S-matrix by S=I+2iT.
(1)
We choose to describe the T-matrix by a sum of complex coupled Breit-Wigners. This implies that the T-matrix is given by
5k
m,1"]j/2 exp(iO 0) 1"}k/zexp(iOlk) m~ - s - i m f ~ o t l ,
(2)
where s is the center-of-mass energy squared, mt is the mass ofte/th resonance and/'tot ~is its total width. O0 is the complex phase of the jth coupling of the lth resonance ( - n < O 0 < n) and 1"0 is the partial width. 1"tot~ is equal to the sum of the partial width. 1"0 is given by
1"o = Pj ~ ,
(3)
where Pj is the phase space factor of the jth channel which is zero below threshold; for the D-wave channels a Blatt-Weisskopfbarrier factor is with radius of one fermi [ 8 ]. Here ~0 represents the coupling of the /th resonance to thejth channel and is a real number. The widths that are given in table 1 are the value of the variable width at s = m 2. In order to describe the J / ~ radiative decay data, we assume each resonance l is produced with a complex production amplitude C~. This implies that the amplitude going to the jth channel from the/th resonance is ~ Ref. [2] includes and cites sources of all the relevant precision data applicable to this analysis. However, to reduce space, the original references to the various data sources are not repeated here.
AO=
Cl~ t/2 r'l/Z exp( iOlj) m] - s - i m t F t o t l
16 January 1992
(4)
We performed a fit to the data of refs. [ 2,5,6,3,7 ] using six S-wave and four D-wave Breit-Wigners with the various parameters including the resonance widths determined by the data fit. One of the S-wave resonances was used to describe the threshold behavior of the nn system with an Adler zero added in the same way as ref. [9]. Another resonance in the Dwave was used to describe the background that is seen above 2.0 GeV in the nn--,KI~ [2] and nn--,qq [5] data. There are 738 degrees of freedom. The y 2 per DOF (degree of freedom) would be ~ 1, if we increased the errors on the average of 20% to allow for systematic errors. Thus the fit is good enough that adding additional poles is not justified. The highest mass S-wave resonance occurs at 2.213 GeV with a total width of 0.366 GeV and is not part of the real focus of this article and thus will not be considered here. Thus this reduces to four S-wave meson resonances and three D-wave meson resonances considered herein. The three D-wave meson resonances are called the t"2(1270), f~( 1525 ) and the f2( 1810) (for some reason this latter state has been omitted from the P D G summary table) [ 10 ]. In the full listings of the Particle Data Group [ 10 ] there are four observations of the t"2( 1810) and thus there appears to be sufficient data to establish it. Only three of the S-wave resonances are well established, the fo(975), fo(1400) and the fo(1720) (which in the P D G summary table is listed as f2 (1720) withj in need of confirmation). Refs. [7,1 ] have shown that the spin of t"2(1720) or 0 (1720) is, with a high probability, 0 and thus should be called the fo(1720). The fo(1720) was first observed in the reaction n - p ~ K s K s n [ 11 ] and confirmed by ref. [ 12] in the same channel. However, it was not associated with the 0(1720) resonance seen in J / ~ radiative decay because of the confusion about the spin of the 0. Now that the 0(1720) spin is to a high probability determined to be 0 [ 7,1 ] it is extremely probable that they are the same object, since within errors the mass and widths are the same. Refs. [ 11,13 ] demonstrate that this spin 0 resonance [ S*' (1720) ] showed the phase motion expected for a resonance, while all t"2(1720) or 0 analyses [ 10 ] have failed to show resonance phase motion. The spin 2 assignment for the 0 also 493
Volume 274, number 3,4
PHYSICS LETTERS B
16 January 1992
Table 1 Parameters of the resonances (see text for details). Resonance 1"2(1270) ~(1525) f2(1810) t"o(975) fo(1400) f0(1720) t"o(1300)
Mass (MeV)
Width (MeV)
F~, (MeV)
X~,
1271_+11 1520_+21 1852_+23 1024_+36 1411_+41 1734_+15 1307_+45
184_+9 134_+11 258_+19 250_+206 260_+27 148_+32 >800
157_+5 8_+4 70_+15 40_+5 134_+7 25_+5
0.85_+0.03 0.06_+0.03 0.27_+0.06 0. l v_o.o6 ~ +°'84
0.52_+0.03 0.17_+0.03 0.11_+0.06
r u n s into p r o b l e m s w h e n o n e looks at the r e a c t i o n K - p - - , K s K s A ° which has a very i m p o r t a n t K-exchange c o m p o n e n t . If the 0 had a 35% KI~ b r a n c h i n g ratio as m e a s u r e d by the M a r k III group [14] it should be p r o d u c e d in K I ( scattering at a sizeable level (see ref. [ 2 ] ) . T h e LASS group m e a s u r e d K - p ~ K~K~A° [ 3 ] a n d did n o t see a n y 0 at all a n d from their f~( 1525 ) one could c o n c l u d e that the 0 m u s t b r a n c h to KI~ at less t h a n 20% [ 15 ]. By switching to a spin 0 the limit o n the KI~ b r a n c h i n g w o u l d increase by (see fig. l a ) . T h i s leads to the r e s o l u t i o n o f the mystery o f the lack o f the 0 in the LASS data. It should be n o t e d that the above d i s c u s s i o n a n d p r e l i m i n a r y fits were p r e s e n t e d by the a u t h o r s at a c o n f e r e n c e [ 16 ] a n d a w o r k s h o p [ 13 ] in 1988.
Fnq
FKK (MeV)
XKI~
21_+ 7 110_+15 7__+ 4 21 ut~+17° _210 118_+19 50_+10
0.11_+0.04 0.82_+0.11 0.03_+0.02 u.~,c~ ~4+o.o6 -0.84 0.45_+0.07 0.34_+0.05 0.11_+0.06
X~q~
(MeV) 6_+2 15_+5 1_+0.7
0.03 _+0.01 0.11 _+0.03 0.014_+0.010
8_+4 18-+4
0.03 _+0.015 0.12 _+0.03 0.43 _+0.26
T h e fourth S-wave r e s o n a n c e fo(1300) which is weakly c o u p l e d to n n a n d KI~ is a n i m p o r t a n t contrib u t i o n to the qq c h a n n e l at t h r e s h o l d (see fig. 2 b ) ; however, o u r fit has a b r o a d width a n d there is little e v i d e n c e of phase m o t i o n with respect to the t"2(1270). This state was first claimed in ref. [ 11 ] a n d there it also d i d n o t show a n y direct phase m o t i o n b u t it had a n a r r o w width due to a r a p i d change in the S-wave m a g n i t u d e . T h i s m a g n i t u d e effect has w e a k e n e d with m o r e m e a s u r e m e n t s in the KI~ channel ~2. T h e fo ( 1 3 0 0 ) could be merely due to u n i t a r i t y effects o n the o p e n i n g o f the qq c h a n n e l . Both T-matrix analyses (ref. [ 11 ] a n d this analysis) have not p u t in u n i t a r i t y since it requires a l o n g - t e r m effort. ~2 Additional data from the experiment quoted in ref. [2].
ISol2 + ID012 i
i
i
i
i
i
I
I
(a)
KK 300
(c) '¥
l. ],/~+/~-
,, KK
~ 7K+K-~
t
'
200 600
100
100
÷S'
200
1.2
1.6
Mass ( KK ) GeV
I
I 1.4 Mass
I
(~n) GeV
I 1.8
I
I
1.4 Mass
I
( KK ) GeV
I
1.8
Fig. 1. (a) The KsKs mass spectrum from the reaction K p~K~K~A° [3] in events as a function of mass in GeV. (b) The J / ~ / r a d i a t i v e decay into y n + n - from ref. [7] in events as a function of mass in GeV. (c) The J/~¢ radiative decay into 7K+K - from ref. [7] in events as a function of mass in GeV. The curve comes from the Breit-Wigner fit described in the text. 494
Volume 274, number 3,4
PHYSICS LETTERS B
ISl2
IDI2 I
I
I
16 January 1992
I
IDI2
I
(a)
I
I
I
I
I
I
I
I
b) xx
," KK
tti
( i
,
,
0.04
0.201 0.03 0.10 0.02
J
1.5
2.0
Mass( Kff) 6eV
1.3
I
2.0 Mass(rlq) GeV
I
1.3
2.0
Mass(-q~)GeV
Fig. 2. (a) The modulus squared of the D-wave in Argand plot units from the reaction nn--, KK, jec= 2 ÷ +, I= 0, obtained from OPE analysis [ 2 ] as a function of mass in GeV. (b) The modulus squared of S-wave in Argand plot units from the reaction nn--,~qrl, jec= 0 ÷÷, I= 0, obtained from OPE analysis [ 5 ] as a function of mass in GeV. (c) The modulus squared of the D-wave in Argand plot units from the reaction nn--*rlq, j.,,c= 2 + ÷, I= 0, obtained from OPE analysis [ 5 ] as a function of mass in GeV. The curve comes from the BreitWigner fit described in the text. However, except near threshold, unitarity effects are not expected to be i m p o r t a n t enough to change the m a j o r conclusions o f this paper. Therefore although practically all calculations o f this type are m o d e l dependent, we believe our m a j o r conclusions will not be affected by the m o d e l used. The resonance fo (1590) o f ref. [ 5 ] is not needed in our analysis even though we fit the qq and rlq' phase shifts that were used to claim the fo(1590) in the first place. In our fit fo(1590) is m a d e up o f a sum o f the fo(1400) and the fo(1720) where it is known from J / ~ radiative decay that the fo(1720) does decay into rpl [ 7 ]. In ref. [ 13 ] we m a d e a fit interchanging fo(1300) with the fo(1590) and obtained a significantly worse fit. It has been argued that the fo(1590) is a new resonance since it is also seen in central production [ 17 ]. First o f all ref. [ 17 ] sees very few events, second one needs to do a simultaneous fit to centrally p r o d u c e d nn, KI~, qrl and 1N' in order to make a convincing statement Third, one needs to explain why the to(1720) is not seen in any o f rl~ data o f refs. [5,17], while it is known that fo(1720) decays into nl] [7] and is p r o d u c e d by nn scattering [2 ] and in central p r o d u c t i o n [ 18 ] because o f its K I ( mode. There is a possibility that the fo(1720) is not p r o d u c e d in central production, because ref. [ 19 ] claims they see a strong 2 + + signal in
the fo(1720) region. At the H a d r o n 89 Conference it was stated that one observes a 2 + + background in the complete mass range of the K~K~ data [ 20 ]. This 2 + ÷ background could be partially due to the t"2(1270) and a2 (1320). However, the spectrum o f ref. [ 20 ] does not show a resonance shape at 1.3 GeV. In table 1 we give the mass, total width, partial widths and the branching fractions (Xj = ~/Ftola I ) for each o f the meson resonances. F o r the fo(1300) we do not give the partial widths but only the branching fractions. Figs. 1-4 show our fit to the data o f refs. [2,5,6,3,7].
3. C o n c l u s i o n
We have p e r f o r m e d a coupled channel analysis on the reactions: nn--,nn, n n ~ K K , n n - , w q , nn--,'qTl' and KI(-~ K_K, where the isoscalar a m p l i t u d e s with j e c = 0 + + and 2 + + were fitted. The channels J / W - ' 7n + n - , J / ~ - - , T K + K - , J/~--qtK~I~, J / w - ' 7 ~ q , and J / ~-~71]i]' were also used in the analysis because they only contain isoscalar mesons with J e C = 0 + + and 2 + + below the mass o f 2.0 GeV. The fits show that the 1"2(1720) or 0 ( 1 7 2 0 ) can be u n d e r s t o o d in form a t i o n p s e u d o s c a l a r - p s e u d o s c a l a r scattering and p r o d u c t i o n J / ~ radiative decays by having a spin 0. 495
Volume 274, number 3,4
PHYSICS LETTERS B
Isl 2 I
16 January 1992
IDI 2
I
I
I
~
I
I
(a)
(b)
I
ISI 2 I
I
I
nn
I
I
I
]
I
. nn
1.2 0.20
0.10 0.4
1.0
1.4 Mass (nn) GeV
v~/IJ
1.8
i I I* ~
1.0
1.4
1.5 2.0 Mass ( KK ) 6eV
1.8
Mass (nn) GeV
Fig. 3. (a) The modulus squared of the S-wave in Argand plot units from the reaction nn~nn, Jet=O++, I=0, obtained from OPE analysis [2] as a function of mass in GeV; the errors (essentially systematic) in the high mass region of the ISI2 mass spectrum are sufficient to explain the systematic displacement of these points. (b) The modulus squared of the D-wave in Argand plot units from the reaction nn--*nn,j e c = 2 + +, I = 0, obtained from OPE analysis [2 ] as a function of mass in GeV. (c) The modulus squared of the S-wave in Argand plot units from the reaction nn--*K((, jpc=0++ 1=0, obtained from OPE analysis [2] as a function of mass in GeV. The curve comes from the Breit-Wigner fit described in the text.
ISt2 0.06
J
T
i
0.03
(a)
200
100
i
i
i
(b)
~P--~ Yqq
t
oo]
i
i
J
(c)
50 P
J
I
I
1.55 1.65 1.75 Mass (qq') GeV
I
1.40 1.80 Mass (qq) GeV
[
1.40
1.70 Mass (qq') GeV
2.00
Fig. 4. (a) The modulus squared of the S-wave in Argand plot units from the reaction nn~qq', JPC=0++, 1=0, obtained from OPE analysis [6] as a function of mass in GeV. (b) The J / ~ radiative decay into 7qq from ref. [7] in events as a function of mass in GeV. (c) The J / ~ radiative decay into ,(ili1' from ref. [7] in events as a function of mass in GeV. The curve comes from a Breit-Wigner fit described in the text.
T h u s in t h i s a n a l y s i s t h e 0 ( 1720 ) w o u l d really b e t h e f o ( 1 7 2 0 ) w h i c h w a s first o b s e r v e d in f o r m a t i o n t h r o u g h nn s c a t t e r i n g b e f o r e it w a s o b s e r v e d in t h e J / r a d i a t i v e decay. I n o u r fit we g e n e r a t e t h e f o ( 1 5 9 0 ) effect d e c a y i n g i n t o q q a n q q ' as a s u m o f t h e fo ( 1 4 0 0 ) 496
a n d f o ( 1 7 2 0 ) in nn s c a t t e r i n g . M o r e d a t a is n e e d e d to u n d e r s t a n d c e n t r a l p r o d u c t i o n . T h e d i f f i c u l t j o b o f a d d i n g u n i t a r i t y to o u r a n a l y s i s is p l a n n e d for t h e fut u r e a n d s h o u l d lead to a b e t t e r u n d e r s t a n d i n g o f t h e f o ( 1 3 0 0 ) a n d t h e qrl t h r e s h o l d .
Volume 274, number 3,4
PHYSICS LETTERS B
References [ 1 ] MK III Collab., L. Chen et al., Amplitude analysis of the KI( system in J / q radiative decay, presented Rheinfels 1990 Workshop on Hadron mass spectrum (St. Goar, FRG, September 1990), preprint SLAC-PUB-5378 (1990). [ 2 ] R.S. Longacre et al., Phys. Lett. B 177 ( 1986 ) 223. [ 3 ] D . Aston et al., in: Proc. BNL Workshop on Glueballs, hybrids, and exotic hadrons (Brookhaven National Laboratory, Upton, NY, August-September 1988), ed. S.U. Chung, AIP Conf. Proc. 185, p. 160. [4] D. Cohen et al., Phys. Rev. D 22 (1980) 2595. [5] D. Aide et al., Nucl. Phys. B 269 (1986) 485. [6] F. Binon et al., Nuovo Cimento A 80 (1984) 363. [ 7 ] T.A. Bolton, Radiative decays of the J / ~ to two pseudoscalar final states, MK Ill Thesis, MIT (April 1988). [8]J. Blatt and V.F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952) p. 361. [9] K.L. Au, D. Morgan and M.R. Pennington, Phys. Rev. D 35 (1987) 1633. [ 10 ] Particle Data Group, J.J. Hermindez et al., Review of particle properties, Phys. Lett. B 239 (1990) 1.
16 January 1992
[ 11 ] A. Etkin et al., Phys. Rev. D 25 (1982) 1786; D 25 (1982) 2446. [ 12] B.V. Bolonkin et al., Nucl. Phys. B 309 (1988) 426. [ 13 ] S.J. Lindenbaum, in: Proc. BNL Workshop on Glueballs, hybrids, and exotic hadrons (Brookhaven National Laboratory, Upton, NY, August-September, 1988 ), ed. S.U. Chung, AlP Conf. Proc. 185, p. 68. [ 14 ] S. Einsweiler, Radiative decays of the • (3097) to two meson final states, MK III Thesis, report SLAC-PUB-272 (1984). [15] R.S. Longacre, in: Proc. BNL Workshop on Glueballs, hybrids and exotic hadrons (Brookhaven National Laboratory, Upton, NY, August-September 1988 ), ed. S.U. Chung, AIP Conf. Proc. 185, p. 670. [16] S.J. Lindenbaum, in: Proc. XXIV Intern. Conf. on High energy physics (Munich, FRG, August, 1988), eds. R. Kotthaus and J.H. Kukn (Springer, Berlin) p. 600. [ 17] D. Aide et al., Phys. Lett. B 201 (1988) 160. [ 18 ] T.A. Armstrong et al., Phys. Lett. B 167 ( 1986 ) 133. [ 19] T.A. Armstrong et al., Phys. Lett. B 227 (1989) 186. [20] T.A. Armstrong et al., in: Proc. Hadron '89 (September, 1988), eds. F. Binon, J.-M. Frere, and J.-P. Peigneux, (Editions Fronti6res, Gif-sur-Yvette, France ) p. 91.
497
P H YSIC S 1 FT T ERS B
C o u p l e d c h a n n e l analysis o f j e c = with m a s s e s b e l o w 2.0 G e V S.J. Lindenbaum a,b and R.S. Longacre
0 + + and 2 + + isoscalar m e s o n s
a
a Brookhaven NationalLaboratory, Upton, N Y 11973, USA b City College of New York, New York, N Y 10031, USA
Received 14 July 1991
A coupled channel analysis of the j e c = 0 ++ and 2 ++ channels below 2 GeV has been performed using Breit-Wigners considering nn scattering amplitudes, J/V radiative decay, and KI~-,KK. The main conclusions were that a good fit with a minimum number of meson poles (four) was obtained in the 0 + ÷ channel. In our fit the fo(1720) was used to describe the first discovered S*' (1720) and the 0 (1720) both of which have the same quantum numbers and parameters to a high probability. Unitarity effects which require a long-termeffort were not included, however they are expected to have important effects primarily near thresholds and thus not affect the major conclusionsof this paper.
1. Introduction
A coupled channel analysis is much more constrained and thus more likely to reveal the m i n i m u m n u m b e r of real resonances required to explain the considerable data available. Therefore, a coupled channel analysis has been performed using BreitWigners considering the n n scattering amplitudes, J / V radiative decay and KI~-,KI~. The spectrum of isoscalar mesons with J P C = O + + and 2 ++ are extracted from pseudoscalar-pseudoscalar final states. It is found that three meson poles are necessary in order to fit the 2 ++ data (1"2(1270), f ' ( 1 5 2 5 ) , and f2 ( 1810) ), while four meson poles are necessary in order to fit the 0 ++ data (fo(975), fo(1300), fo(1400), and f0(1720) ) up to 2.0 GeV. This analysis uses the fact that the f2(1720) or 0 ( 1 7 2 0 ) that is observed in the J/V radiative decay has spin 0, not spin 2. This was found by the M K III Collaboration to be highly probable [ 1 ]. Earlier fits [2 ] (which are still applicable) with the 0 having spin 2 did not agree with the LASS data [3]. A fit was achieved without the fo(1590) while the phase shifts This research was supported by the US Department of Energy under Contract Nos. DE-AC02-76CH00016 and DE-AC0283ER40107. 492
which lead to the claim for this resonance are well described using the fo(1300), fo(1400), and the fo(1720). The 1"o(1300) has a large width and does not show sufficient phase motion, thus it could possibly be due to an unitary effect at the rlq threshold. In our fit the fo(1590) is made up of a sum of the fo(1400) and the fo(1720).
2. Coupled channel analysis of j P C = 0 + + and 2 ÷ ÷
When a pion strikes a proton thereby changing it into a neutron, the low m o m e n t u m transfer It' I scattering is d o m i n a t e d by one pion exchange (OPE). Using the standard OPE extrapolation [4], one obtains the S-matrix Argand amplitudes for each of the partial waves. The spectrum of i soscalar mesons in the j p c = 0 + + and 2 + + channels is very important in testing q u a r k gluon and g l u o n - g l u o n confined Q C D theory. Both qdl and gg states can occur in these channels. However, they can mix, and therefore only a detailed analysis of these mesons in different production and decay channels can determine their nature.
0370-2693/92/$ 05.00 © 1992 Elsevier SciencePublishers B.V. All rights reserved.
Volume 274, number 3,4
PHYSICS LETTERS B
For the following S-matrix analysis we have used the processes nn--,nn, nn--,KK, nn--,qq [2,5] ~, nn--,~n' [6]. We also added the reaction KI~--,KI~ [3] by assuming that the unnatural exchange for M = 0 amplitude for K - p - + ~ ° [3] is pure K exchange. The channels J / t ~ T n + n -, J/~--,TK+K -, J/~-q,K~I~, J/~c--,Tqrl, and J/w-,yqrl' [7] were also used in the S-matrix analysis because they only contain isoscalar mesons with jpc= 0 + + and 2 + + below the mass of 2.0 GeV. We proceed to obtain the S-matrix, which describes meson-meson scattering for the quantum numbers I a = 0 + , JPC=O++ and 2 ++. In particular we concentrate on the pseudoscalar-pseudoscalar couplings. The T-matrix is related to the S-matrix by S=I+2iT.
(1)
We choose to describe the T-matrix by a sum of complex coupled Breit-Wigners. This implies that the T-matrix is given by
5k
m,1"]j/2 exp(iO 0) 1"}k/zexp(iOlk) m~ - s - i m f ~ o t l ,
(2)
where s is the center-of-mass energy squared, mt is the mass ofte/th resonance and/'tot ~is its total width. O0 is the complex phase of the jth coupling of the lth resonance ( - n < O 0 < n) and 1"0 is the partial width. 1"tot~ is equal to the sum of the partial width. 1"0 is given by
1"o = Pj ~ ,
(3)
where Pj is the phase space factor of the jth channel which is zero below threshold; for the D-wave channels a Blatt-Weisskopfbarrier factor is with radius of one fermi [ 8 ]. Here ~0 represents the coupling of the /th resonance to thejth channel and is a real number. The widths that are given in table 1 are the value of the variable width at s = m 2. In order to describe the J / ~ radiative decay data, we assume each resonance l is produced with a complex production amplitude C~. This implies that the amplitude going to the jth channel from the/th resonance is ~ Ref. [2] includes and cites sources of all the relevant precision data applicable to this analysis. However, to reduce space, the original references to the various data sources are not repeated here.
AO=
Cl~ t/2 r'l/Z exp( iOlj) m] - s - i m t F t o t l
16 January 1992
(4)
We performed a fit to the data of refs. [ 2,5,6,3,7 ] using six S-wave and four D-wave Breit-Wigners with the various parameters including the resonance widths determined by the data fit. One of the S-wave resonances was used to describe the threshold behavior of the nn system with an Adler zero added in the same way as ref. [9]. Another resonance in the Dwave was used to describe the background that is seen above 2.0 GeV in the nn--,KI~ [2] and nn--,qq [5] data. There are 738 degrees of freedom. The y 2 per DOF (degree of freedom) would be ~ 1, if we increased the errors on the average of 20% to allow for systematic errors. Thus the fit is good enough that adding additional poles is not justified. The highest mass S-wave resonance occurs at 2.213 GeV with a total width of 0.366 GeV and is not part of the real focus of this article and thus will not be considered here. Thus this reduces to four S-wave meson resonances and three D-wave meson resonances considered herein. The three D-wave meson resonances are called the t"2(1270), f~( 1525 ) and the f2( 1810) (for some reason this latter state has been omitted from the P D G summary table) [ 10 ]. In the full listings of the Particle Data Group [ 10 ] there are four observations of the t"2( 1810) and thus there appears to be sufficient data to establish it. Only three of the S-wave resonances are well established, the fo(975), fo(1400) and the fo(1720) (which in the P D G summary table is listed as f2 (1720) withj in need of confirmation). Refs. [7,1 ] have shown that the spin of t"2(1720) or 0 (1720) is, with a high probability, 0 and thus should be called the fo(1720). The fo(1720) was first observed in the reaction n - p ~ K s K s n [ 11 ] and confirmed by ref. [ 12] in the same channel. However, it was not associated with the 0(1720) resonance seen in J / ~ radiative decay because of the confusion about the spin of the 0. Now that the 0(1720) spin is to a high probability determined to be 0 [ 7,1 ] it is extremely probable that they are the same object, since within errors the mass and widths are the same. Refs. [ 11,13 ] demonstrate that this spin 0 resonance [ S*' (1720) ] showed the phase motion expected for a resonance, while all t"2(1720) or 0 analyses [ 10 ] have failed to show resonance phase motion. The spin 2 assignment for the 0 also 493
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PHYSICS LETTERS B
16 January 1992
Table 1 Parameters of the resonances (see text for details). Resonance 1"2(1270) ~(1525) f2(1810) t"o(975) fo(1400) f0(1720) t"o(1300)
Mass (MeV)
Width (MeV)
F~, (MeV)
X~,
1271_+11 1520_+21 1852_+23 1024_+36 1411_+41 1734_+15 1307_+45
184_+9 134_+11 258_+19 250_+206 260_+27 148_+32 >800
157_+5 8_+4 70_+15 40_+5 134_+7 25_+5
0.85_+0.03 0.06_+0.03 0.27_+0.06 0. l v_o.o6 ~ +°'84
0.52_+0.03 0.17_+0.03 0.11_+0.06
r u n s into p r o b l e m s w h e n o n e looks at the r e a c t i o n K - p - - , K s K s A ° which has a very i m p o r t a n t K-exchange c o m p o n e n t . If the 0 had a 35% KI~ b r a n c h i n g ratio as m e a s u r e d by the M a r k III group [14] it should be p r o d u c e d in K I ( scattering at a sizeable level (see ref. [ 2 ] ) . T h e LASS group m e a s u r e d K - p ~ K~K~A° [ 3 ] a n d did n o t see a n y 0 at all a n d from their f~( 1525 ) one could c o n c l u d e that the 0 m u s t b r a n c h to KI~ at less t h a n 20% [ 15 ]. By switching to a spin 0 the limit o n the KI~ b r a n c h i n g w o u l d increase by (see fig. l a ) . T h i s leads to the r e s o l u t i o n o f the mystery o f the lack o f the 0 in the LASS data. It should be n o t e d that the above d i s c u s s i o n a n d p r e l i m i n a r y fits were p r e s e n t e d by the a u t h o r s at a c o n f e r e n c e [ 16 ] a n d a w o r k s h o p [ 13 ] in 1988.
Fnq
FKK (MeV)
XKI~
21_+ 7 110_+15 7__+ 4 21 ut~+17° _210 118_+19 50_+10
0.11_+0.04 0.82_+0.11 0.03_+0.02 u.~,c~ ~4+o.o6 -0.84 0.45_+0.07 0.34_+0.05 0.11_+0.06
X~q~
(MeV) 6_+2 15_+5 1_+0.7
0.03 _+0.01 0.11 _+0.03 0.014_+0.010
8_+4 18-+4
0.03 _+0.015 0.12 _+0.03 0.43 _+0.26
T h e fourth S-wave r e s o n a n c e fo(1300) which is weakly c o u p l e d to n n a n d KI~ is a n i m p o r t a n t contrib u t i o n to the qq c h a n n e l at t h r e s h o l d (see fig. 2 b ) ; however, o u r fit has a b r o a d width a n d there is little e v i d e n c e of phase m o t i o n with respect to the t"2(1270). This state was first claimed in ref. [ 11 ] a n d there it also d i d n o t show a n y direct phase m o t i o n b u t it had a n a r r o w width due to a r a p i d change in the S-wave m a g n i t u d e . T h i s m a g n i t u d e effect has w e a k e n e d with m o r e m e a s u r e m e n t s in the KI~ channel ~2. T h e fo ( 1 3 0 0 ) could be merely due to u n i t a r i t y effects o n the o p e n i n g o f the qq c h a n n e l . Both T-matrix analyses (ref. [ 11 ] a n d this analysis) have not p u t in u n i t a r i t y since it requires a l o n g - t e r m effort. ~2 Additional data from the experiment quoted in ref. [2].
ISol2 + ID012 i
i
i
i
i
i
I
I
(a)
KK 300
(c) '¥
l. ],/~+/~-
,, KK
~ 7K+K-~
t
'
200 600
100
100
÷S'
200
1.2
1.6
Mass ( KK ) GeV
I
I 1.4 Mass
I
(~n) GeV
I 1.8
I
I
1.4 Mass
I
( KK ) GeV
I
1.8
Fig. 1. (a) The KsKs mass spectrum from the reaction K p~K~K~A° [3] in events as a function of mass in GeV. (b) The J / ~ / r a d i a t i v e decay into y n + n - from ref. [7] in events as a function of mass in GeV. (c) The J/~¢ radiative decay into 7K+K - from ref. [7] in events as a function of mass in GeV. The curve comes from the Breit-Wigner fit described in the text. 494
Volume 274, number 3,4
PHYSICS LETTERS B
ISl2
IDI2 I
I
I
16 January 1992
I
IDI2
I
(a)
I
I
I
I
I
I
I
I
b) xx
," KK
tti
( i
,
,
0.04
0.201 0.03 0.10 0.02
J
1.5
2.0
Mass( Kff) 6eV
1.3
I
2.0 Mass(rlq) GeV
I
1.3
2.0
Mass(-q~)GeV
Fig. 2. (a) The modulus squared of the D-wave in Argand plot units from the reaction nn--, KK, jec= 2 ÷ +, I= 0, obtained from OPE analysis [ 2 ] as a function of mass in GeV. (b) The modulus squared of S-wave in Argand plot units from the reaction nn--,~qrl, jec= 0 ÷÷, I= 0, obtained from OPE analysis [ 5 ] as a function of mass in GeV. (c) The modulus squared of the D-wave in Argand plot units from the reaction nn--*rlq, j.,,c= 2 + ÷, I= 0, obtained from OPE analysis [ 5 ] as a function of mass in GeV. The curve comes from the BreitWigner fit described in the text. However, except near threshold, unitarity effects are not expected to be i m p o r t a n t enough to change the m a j o r conclusions o f this paper. Therefore although practically all calculations o f this type are m o d e l dependent, we believe our m a j o r conclusions will not be affected by the m o d e l used. The resonance fo (1590) o f ref. [ 5 ] is not needed in our analysis even though we fit the qq and rlq' phase shifts that were used to claim the fo(1590) in the first place. In our fit fo(1590) is m a d e up o f a sum o f the fo(1400) and the fo(1720) where it is known from J / ~ radiative decay that the fo(1720) does decay into rpl [ 7 ]. In ref. [ 13 ] we m a d e a fit interchanging fo(1300) with the fo(1590) and obtained a significantly worse fit. It has been argued that the fo(1590) is a new resonance since it is also seen in central production [ 17 ]. First o f all ref. [ 17 ] sees very few events, second one needs to do a simultaneous fit to centrally p r o d u c e d nn, KI~, qrl and 1N' in order to make a convincing statement Third, one needs to explain why the to(1720) is not seen in any o f rl~ data o f refs. [5,17], while it is known that fo(1720) decays into nl] [7] and is p r o d u c e d by nn scattering [2 ] and in central p r o d u c t i o n [ 18 ] because o f its K I ( mode. There is a possibility that the fo(1720) is not p r o d u c e d in central production, because ref. [ 19 ] claims they see a strong 2 + + signal in
the fo(1720) region. At the H a d r o n 89 Conference it was stated that one observes a 2 + + background in the complete mass range of the K~K~ data [ 20 ]. This 2 + ÷ background could be partially due to the t"2(1270) and a2 (1320). However, the spectrum o f ref. [ 20 ] does not show a resonance shape at 1.3 GeV. In table 1 we give the mass, total width, partial widths and the branching fractions (Xj = ~/Ftola I ) for each o f the meson resonances. F o r the fo(1300) we do not give the partial widths but only the branching fractions. Figs. 1-4 show our fit to the data o f refs. [2,5,6,3,7].
3. C o n c l u s i o n
We have p e r f o r m e d a coupled channel analysis on the reactions: nn--,nn, n n ~ K K , n n - , w q , nn--,'qTl' and KI(-~ K_K, where the isoscalar a m p l i t u d e s with j e c = 0 + + and 2 + + were fitted. The channels J / W - ' 7n + n - , J / ~ - - , T K + K - , J/~--qtK~I~, J / w - ' 7 ~ q , and J / ~-~71]i]' were also used in the analysis because they only contain isoscalar mesons with J e C = 0 + + and 2 + + below the mass o f 2.0 GeV. The fits show that the 1"2(1720) or 0 ( 1 7 2 0 ) can be u n d e r s t o o d in form a t i o n p s e u d o s c a l a r - p s e u d o s c a l a r scattering and p r o d u c t i o n J / ~ radiative decays by having a spin 0. 495
Volume 274, number 3,4
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Isl 2 I
16 January 1992
IDI 2
I
I
I
~
I
I
(a)
(b)
I
ISI 2 I
I
I
nn
I
I
I
]
I
. nn
1.2 0.20
0.10 0.4
1.0
1.4 Mass (nn) GeV
v~/IJ
1.8
i I I* ~
1.0
1.4
1.5 2.0 Mass ( KK ) 6eV
1.8
Mass (nn) GeV
Fig. 3. (a) The modulus squared of the S-wave in Argand plot units from the reaction nn~nn, Jet=O++, I=0, obtained from OPE analysis [2] as a function of mass in GeV; the errors (essentially systematic) in the high mass region of the ISI2 mass spectrum are sufficient to explain the systematic displacement of these points. (b) The modulus squared of the D-wave in Argand plot units from the reaction nn--*nn,j e c = 2 + +, I = 0, obtained from OPE analysis [2 ] as a function of mass in GeV. (c) The modulus squared of the S-wave in Argand plot units from the reaction nn--*K((, jpc=0++ 1=0, obtained from OPE analysis [2] as a function of mass in GeV. The curve comes from the Breit-Wigner fit described in the text.
ISt2 0.06
J
T
i
0.03
(a)
200
100
i
i
i
(b)
~P--~ Yqq
t
oo]
i
i
J
(c)
50 P
J
I
I
1.55 1.65 1.75 Mass (qq') GeV
I
1.40 1.80 Mass (qq) GeV
[
1.40
1.70 Mass (qq') GeV
2.00
Fig. 4. (a) The modulus squared of the S-wave in Argand plot units from the reaction nn~qq', JPC=0++, 1=0, obtained from OPE analysis [6] as a function of mass in GeV. (b) The J / ~ radiative decay into 7qq from ref. [7] in events as a function of mass in GeV. (c) The J / ~ radiative decay into ,(ili1' from ref. [7] in events as a function of mass in GeV. The curve comes from a Breit-Wigner fit described in the text.
T h u s in t h i s a n a l y s i s t h e 0 ( 1720 ) w o u l d really b e t h e f o ( 1 7 2 0 ) w h i c h w a s first o b s e r v e d in f o r m a t i o n t h r o u g h nn s c a t t e r i n g b e f o r e it w a s o b s e r v e d in t h e J / r a d i a t i v e decay. I n o u r fit we g e n e r a t e t h e f o ( 1 5 9 0 ) effect d e c a y i n g i n t o q q a n q q ' as a s u m o f t h e fo ( 1 4 0 0 ) 496
a n d f o ( 1 7 2 0 ) in nn s c a t t e r i n g . M o r e d a t a is n e e d e d to u n d e r s t a n d c e n t r a l p r o d u c t i o n . T h e d i f f i c u l t j o b o f a d d i n g u n i t a r i t y to o u r a n a l y s i s is p l a n n e d for t h e fut u r e a n d s h o u l d lead to a b e t t e r u n d e r s t a n d i n g o f t h e f o ( 1 3 0 0 ) a n d t h e qrl t h r e s h o l d .
Volume 274, number 3,4
PHYSICS LETTERS B
References [ 1 ] MK III Collab., L. Chen et al., Amplitude analysis of the KI( system in J / q radiative decay, presented Rheinfels 1990 Workshop on Hadron mass spectrum (St. Goar, FRG, September 1990), preprint SLAC-PUB-5378 (1990). [ 2 ] R.S. Longacre et al., Phys. Lett. B 177 ( 1986 ) 223. [ 3 ] D . Aston et al., in: Proc. BNL Workshop on Glueballs, hybrids, and exotic hadrons (Brookhaven National Laboratory, Upton, NY, August-September 1988), ed. S.U. Chung, AIP Conf. Proc. 185, p. 160. [4] D. Cohen et al., Phys. Rev. D 22 (1980) 2595. [5] D. Aide et al., Nucl. Phys. B 269 (1986) 485. [6] F. Binon et al., Nuovo Cimento A 80 (1984) 363. [ 7 ] T.A. Bolton, Radiative decays of the J / ~ to two pseudoscalar final states, MK Ill Thesis, MIT (April 1988). [8]J. Blatt and V.F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952) p. 361. [9] K.L. Au, D. Morgan and M.R. Pennington, Phys. Rev. D 35 (1987) 1633. [ 10 ] Particle Data Group, J.J. Hermindez et al., Review of particle properties, Phys. Lett. B 239 (1990) 1.
16 January 1992
[ 11 ] A. Etkin et al., Phys. Rev. D 25 (1982) 1786; D 25 (1982) 2446. [ 12] B.V. Bolonkin et al., Nucl. Phys. B 309 (1988) 426. [ 13 ] S.J. Lindenbaum, in: Proc. BNL Workshop on Glueballs, hybrids, and exotic hadrons (Brookhaven National Laboratory, Upton, NY, August-September, 1988 ), ed. S.U. Chung, AlP Conf. Proc. 185, p. 68. [ 14 ] S. Einsweiler, Radiative decays of the • (3097) to two meson final states, MK III Thesis, report SLAC-PUB-272 (1984). [15] R.S. Longacre, in: Proc. BNL Workshop on Glueballs, hybrids and exotic hadrons (Brookhaven National Laboratory, Upton, NY, August-September 1988 ), ed. S.U. Chung, AIP Conf. Proc. 185, p. 670. [16] S.J. Lindenbaum, in: Proc. XXIV Intern. Conf. on High energy physics (Munich, FRG, August, 1988), eds. R. Kotthaus and J.H. Kukn (Springer, Berlin) p. 600. [ 17] D. Aide et al., Phys. Lett. B 201 (1988) 160. [ 18 ] T.A. Armstrong et al., Phys. Lett. B 167 ( 1986 ) 133. [ 19] T.A. Armstrong et al., Phys. Lett. B 227 (1989) 186. [20] T.A. Armstrong et al., in: Proc. Hadron '89 (September, 1988), eds. F. Binon, J.-M. Frere, and J.-P. Peigneux, (Editions Fronti6res, Gif-sur-Yvette, France ) p. 91.
497