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Quarkonia spectra from lattice gauge theories
Volume 147B, number 4,5
PHYSICS LETTERS
8 November 1984
Q U A R K O N I A SPECTRA F R O M L A T I ' I C E G A U G E T H E O R I E S Massimo C A M P O S T R I N I Scuola Normale Superiore, 1-56100 Pisa1, Italy INFN Sezione di Pisa, 1-56100 Pisa, Italy and Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
Received 5 July 1984
A quark-antiquark potential obtained from a Monte Carlo simulation for the pure SU(3) lattice gauge theory is modified for light quark effects and used to calculate charmonium and beautonium spectra. Results are in good agreement with experimental data.
1. Introduction. Heavy quark-antiquark bound states ("quarkonia") are believed to be the simplest hadronic systems, the "hydrogen atom" where q u a n t u m chromodynamics can be tested. The ff and T family masses are well predicted by various non-relativistic models in terms of a flavour-independent potential [1-3]. However, f r o m the Q C D point of view, these potentials are at best an interpolation between the small distance perturbative one- or two-gluon exchange and the long range linear confining potential, largely determined b y fits and comparison with experimental data and with no relation to the basic gauge theory. Moreover, some successful potentials do not agree with these long and short distance limits, so one cannot say that Q C D is indeed proven in these models [4]. In fact, quarkonia levels are sensitive to the potential in the region f r o m about 0.1 to 1 fm, where neither limit works. A better probe of the short distance behaviour of the potential is the leptonic width F ( e + e - ) , whose theoretical calculations have, however, larger uncertainties [1].
lattice, where the force between two static sources can be directly measured [5,6]. Due to mainly numerical difficulties, no data are at present available for the full lattice QCD, with dynamical fermions. In ref. [6] a high statistic Monte Carlo simulation is performed on a large lattice for the pure SU(3) gauge theory, and the force is determined f r o m Wilson loop values. In particular, an analytical formula is obtained that fits the Monte Carlo data very well. Defining a ( r ) according to F ( r ) = ~ a ( r ) / r 2,
(2.1)
the renormalization group equation satisfied by a ( r ) is: d a / d log ( r 2) = f ( a ) ,
(2.2)
with
f(a) = (fo/4 ~r)a 2 + [f1/(4~r)2]a3 + O(a 4),
(2.3) 2. The potential. The only present method to
at small a. A constant force at large r implies
obtain a q u a r k - a n t i q u a r k potential directly f r o m Q C D is a Monte Carlo simulation on a
f ( a ) o~ a for a ---, oo. A rational approximation is
1Address after August 1984: ScuolaNormale Superiore, Piazza dei Cavalieri 7, 1-56100 Pisa, Italy
f(a)--a2[(a + ao)2 + a~]/Q(a),
so chosen for f :
0 3 7 0 - 2 6 9 3 / 8 4 / $ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2.4) 343
Volume 147B, number 4,5
PHYSICS LETTERS
with Q(a) third degree polynomial. After integration, we obtain log (rA¢)2
_
0
_ Cola _ C, log ( a/C2 )
+ C 310g{l(a + ao)2+ a2l/(a 2 + a 2)} -I- C 4
8 November 1984
o"
[tan - l ( ( a + O~0)/Otl) -- tan -1(Ot0/O~l)] 0
(2.5) where A~ is the scale parameter related to the short distance behavior of F(r) [5,7]. Imposing F(r) = e + O ( 1 / r ) at large r and eq. (2.3):
O"
u~ i
Co=C2=4*r/fo, Cl=fl/fo 2, C 3 = 1 ( 1 + C 1 ) , C4 = (log( ~CoA2jo)-C310g[C2/(a 2 + a2)]} × [½,r- t a n - l ( a 0 / a l ) ] - 1
(2.6)
From perturbation theory we know f0 = 11 - 2Nf,
fl = 102 - ~Nf,
I
0.0
(2.7)
( N f = 0),
Ac = 49.03A w
(Nf = 3),
(2.8)
where A w is the scale parameter of the lattice Wilson action. Setting N F = 0, A 2 / o and a0, a 1 are determined by fits to the Monte Carlo data: A w = 9.24 x 10-37rff, a 0 = 0.084,
a 1 = 0.202
(2.9)
(see ref. [6] for details). This potential looks rather different from successful phenomenological ones (see fig. 1) and indeed it gives inaccurate values of quarkonia masses. We of course ascribe this discrepancy to light quarks effects. The success of the quenched approximation in estimating the masses of the lightest hadrons [8] tells us that low energy quark loops contributions are relatively small; therefore, we approximate 344
.
.
.
I
1.0 r
.
.
.
.
I
1.5
(fm)
Fig. 1. Solid lines: Monte Carlo (upper) and modified (lower) potentials. Dashed and dotted lines: the potentials of refs. [2] and [3], respectively, shifted to coincide with the modified potential for r = 0.5 fm.
and, from ref. [7]: Ac = 30.19Aw
.
0•5
fermionic contributions at the higher energies involved in quarkonia physics with their perturbative calculation. So we retain the form (2.5), (2.6) for the potential and the values (2.9) for A2w/O, a o and al, and take for f0, fl and A J A w the values (2.7), (2.8) for Nf = 3. The Monte Carlo potential and this modification are shown in fig. 1, together with two sample successful phenomenological potentials; other potentials are very similar to these in the range 0.1-1 fm (see ref. [1]). Given the string tension o, the potential is fully determined, except for an additive constant. This constant could in principle be determined by Monte Carlo calculations, e.g. by directly measuring bound state masses in the quenched approximation. Work is in progress in trying to extract this constant from Wilson loop values only.
3. Results• We use the phenomenological value ¢~- = 420 MeV, obtained from the Regge slope of
Volume 147B, number 4,5
8 November 1984
PHYSICS LETTERS
Table 1 ~k family for m c = 1.70 GeV; experimental data are taken from ref. [11]. Numbers in brackets are from experiment. State
Mass (GeV)
r(e + e - ) / r ( l S -, e + e-)a)
(t)2/e
1S 1P 2S 1D 2P 3S 2D 3P 4S 3D 5S
3.10 3.51 3.69 3.80 3.96 4.11 4.19 4.34 4.48 4.53 4.80
1.00
0.20 0.22 0.26 0.25 0.29 0.32 0.32 0.35 0.38 0.38 0.43
(3.10) (3.50) (3.69) (3.77)
0.48 (0.41 + 0.15)
(4.03 + 0.05) (4.16_+ 0.02)
0.34 ( - 0.2)
(4.42_+ 0.01)
0.26 0.22
2>
(( r 2 ))1/2 (fm)
0.40 0.62 0.79 0.81 0.97 1.12 1.13 1.27 1.40 1.41 1.65
a)F(1S --, e+e - ) = 6.17 keV (experiment: 4.6 _+ 1.4 keV).
t h e p t r a j e c t o r y . T h i s gives, u s i n g (2.9), t h e v a l u e
[71
(3.1)
A~ss = 50A w = 2 0 0 M e V
( c o m p a r e w i t h ref. [9]). A l l t h e r e s u l t s are fairly i n s e n s i t i v e to a 100 M e V u n c e r t a i n t y in t h e q u a r k m a s s e s ( c o m p e n s a t e d b y a n a p p r o p r i a t e shift in the a d d i t i v e c o n s t a n t ) , so w e c a n e s t i m a t e t h e m f r o m e x p e r i m e n t a l d a t a . W e t a k e for t h e c h a r m q u a r k m a s s
m c = m o - 150 M e V = 1700 M e V .
(3.2)
T h e a d d i t i v e c o n s t a n t a n d the b e a u t y q u a r k m a s s a r e t h e n o b t a i n e d b y fixing t h e m a s s e s of t h e g r o u n d s t a t e s ~k a n d 7' to t h e c o r r e c t e x p e r i m e n t a l v a l u e s . By n u m e r i c a l c a l c u l a t i o n , w e o b t a i n t h e r e s u l t s s h o w n i n t a b l e s 1 a n d 2; the l e p t o n i c w i d t h s a r e c a l c u l a t e d via [10]
F(nS---, e+e -) = { 16~e2qa2/[m(nS)l 2} × [1 - ( 1 6 / 3 ¢ r ) a s ( 2 m q )
] [ ~ , s ( 0 ) 12.
(3.3)
R e l a t e d u n c e r t a i n t i e s are d i s c u s s e d in ref. [1]. T h e m a x i m u m l e v e l shifts for a 100 M e V v a r i a t i o n in m q a r e 20 M e V f o r t h e ~k a n d 3 M e V f o r t h e T
Table 2 T family for m b = 5.07 GeV; experimental data are taken from refs. [12] and [13] (5S state). Numbers in brackets are from experiment. State
Mass (GeV)
r(e + e - ) / r ( l S --, e + e- )a)
( U2/C 2 )
( ( r 2 ))1/2 (fm)
1S 1P 2S 1D 2P 3S 2D 3P 4S 4P 5S
9.46 9.87 9.99 10.11 10.22 10.33 10.41 10.51 10.60 10.75 10.84
1.00
0.070 0.064 0.072 0.068 0.076 0.083 0.081 0.088 0.095 0.099 0.107
0.23 0.39 0.50 0.53 0.64 0.73 0.75 0.85 0.94 1.04 1.11
(9.46) (9.90) (10.02)
0.47
(0.42 _+0.02)
(10.26) (10.36)
0.35
(0.31 +_0.02)
(10.58)
0.29 (0.24 + 0.03)
(10.87)
0.26
a)F(1S --" e + e - ) = 1.17 keV (experiment: 1.22 +_0.03 keV). 345
Volume 147B, number 4,5
PHYSICS LETTERS
f a m i l i e s ; o t h e r q u a n t i t i e s are e v e n m o r e stable. A 10 M e V v a r i a t i o n i n x/b- gives a level shift u p to 30 M e V a n d a 5% v a r i a t i o n i n F ( e ÷ e - ) for the two families. T h e s e results c o m p a r e well with experim e n t a l d a t a (see ref. [11], a n d ref. [12] for the T f a m i l y ) : t h e m a s s differences are correct w i t h i n the e x p e c t e d - 5% relativistic c o r r e c t i o n s for the T a n d e v e n b e t t e r for the ~k families; T l e p t o n i c w i d t h s are also correct w i t h i n the e s t i m a t e d 10% c o r r e c t i o n s [1]. F i n a l l y , f r o m the b e a u t y q u a r k m a s s we o b t a i n rn B = m D - m c + m b = 5.24 G e V .
(3.4)
I w i s h to t h a n k C. R e b b i for p r o v i d i n g his r e s u l t s a n d for m a n y h e l p f u l discussions. T h i s w o r k was p a r t i a l l y s u p p o r t e d b y the D e l l a R i c c i a Foundation.
References [1] W. Buchmialler and S.-H.H. Tye, Phys. Rev. D24 (1981) 132, and references therein.
346
8 November 1984
[2] E. Eichten et al., Phys. Rev. D17 (1978) 3090; D21 (1980) 203. [3] G. Bhanot and S. Rudaz, Phys. Lett. 78B (1978) 119. [4] A. Martin, Phys. Lett. 93B (1980) 338. [5] M. Creutz, Proc. Fourth Johns Hopkins Workshop on Current problems in particle theory (Bonn, 1980); J.D. Stack, Phys. Rev. D27 (1983) 412; ITP-Santa Barbara preprint No. 136 (1983). [6] D. Barkai, K.J.M. Moriarty and C. Rebbi, Brookhaven preprint BNL-34923 (1984). [7] A. Billoire, Phys. Lett. 104B (1981) 472. [8] H. Hamber and G. Parisi, Phys. Lett. 47 (1981) 1792; Phys. Rev. D27 (1983) 208; E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795; F. Fucito et al., Nucl. Phys. B210[FS6] (1982) 407; D. Weingarten, Nucl. Phys. B215[FS7] (1983) 1; C. Bernard, T. Draper and K. Olnyk, Phys. Rev. D27 (1983) 227. [9] B. Adeva et al., Phys. Rev. Lett. 50 (1983) 2051. [10] R. Barbieri et al., Phys. Lett. 57B (1975) 455. [11] Particle Data Group, Phys. Lett. 111B (1983) 1. [12] P. Franzini, Columbia University preprint Nevis R # 1300 (1984). [13] P. Kass, talk XIXth Rencontre de Moriond, New particle production at high energies (March 1984).
PHYSICS LETTERS
8 November 1984
Q U A R K O N I A SPECTRA F R O M L A T I ' I C E G A U G E T H E O R I E S Massimo C A M P O S T R I N I Scuola Normale Superiore, 1-56100 Pisa1, Italy INFN Sezione di Pisa, 1-56100 Pisa, Italy and Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
Received 5 July 1984
A quark-antiquark potential obtained from a Monte Carlo simulation for the pure SU(3) lattice gauge theory is modified for light quark effects and used to calculate charmonium and beautonium spectra. Results are in good agreement with experimental data.
1. Introduction. Heavy quark-antiquark bound states ("quarkonia") are believed to be the simplest hadronic systems, the "hydrogen atom" where q u a n t u m chromodynamics can be tested. The ff and T family masses are well predicted by various non-relativistic models in terms of a flavour-independent potential [1-3]. However, f r o m the Q C D point of view, these potentials are at best an interpolation between the small distance perturbative one- or two-gluon exchange and the long range linear confining potential, largely determined b y fits and comparison with experimental data and with no relation to the basic gauge theory. Moreover, some successful potentials do not agree with these long and short distance limits, so one cannot say that Q C D is indeed proven in these models [4]. In fact, quarkonia levels are sensitive to the potential in the region f r o m about 0.1 to 1 fm, where neither limit works. A better probe of the short distance behaviour of the potential is the leptonic width F ( e + e - ) , whose theoretical calculations have, however, larger uncertainties [1].
lattice, where the force between two static sources can be directly measured [5,6]. Due to mainly numerical difficulties, no data are at present available for the full lattice QCD, with dynamical fermions. In ref. [6] a high statistic Monte Carlo simulation is performed on a large lattice for the pure SU(3) gauge theory, and the force is determined f r o m Wilson loop values. In particular, an analytical formula is obtained that fits the Monte Carlo data very well. Defining a ( r ) according to F ( r ) = ~ a ( r ) / r 2,
(2.1)
the renormalization group equation satisfied by a ( r ) is: d a / d log ( r 2) = f ( a ) ,
(2.2)
with
f(a) = (fo/4 ~r)a 2 + [f1/(4~r)2]a3 + O(a 4),
(2.3) 2. The potential. The only present method to
at small a. A constant force at large r implies
obtain a q u a r k - a n t i q u a r k potential directly f r o m Q C D is a Monte Carlo simulation on a
f ( a ) o~ a for a ---, oo. A rational approximation is
1Address after August 1984: ScuolaNormale Superiore, Piazza dei Cavalieri 7, 1-56100 Pisa, Italy
f(a)--a2[(a + ao)2 + a~]/Q(a),
so chosen for f :
0 3 7 0 - 2 6 9 3 / 8 4 / $ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2.4) 343
Volume 147B, number 4,5
PHYSICS LETTERS
with Q(a) third degree polynomial. After integration, we obtain log (rA¢)2
_
0
_ Cola _ C, log ( a/C2 )
+ C 310g{l(a + ao)2+ a2l/(a 2 + a 2)} -I- C 4
8 November 1984
o"
[tan - l ( ( a + O~0)/Otl) -- tan -1(Ot0/O~l)] 0
(2.5) where A~ is the scale parameter related to the short distance behavior of F(r) [5,7]. Imposing F(r) = e + O ( 1 / r ) at large r and eq. (2.3):
O"
u~ i
Co=C2=4*r/fo, Cl=fl/fo 2, C 3 = 1 ( 1 + C 1 ) , C4 = (log( ~CoA2jo)-C310g[C2/(a 2 + a2)]} × [½,r- t a n - l ( a 0 / a l ) ] - 1
(2.6)
From perturbation theory we know f0 = 11 - 2Nf,
fl = 102 - ~Nf,
I
0.0
(2.7)
( N f = 0),
Ac = 49.03A w
(Nf = 3),
(2.8)
where A w is the scale parameter of the lattice Wilson action. Setting N F = 0, A 2 / o and a0, a 1 are determined by fits to the Monte Carlo data: A w = 9.24 x 10-37rff, a 0 = 0.084,
a 1 = 0.202
(2.9)
(see ref. [6] for details). This potential looks rather different from successful phenomenological ones (see fig. 1) and indeed it gives inaccurate values of quarkonia masses. We of course ascribe this discrepancy to light quarks effects. The success of the quenched approximation in estimating the masses of the lightest hadrons [8] tells us that low energy quark loops contributions are relatively small; therefore, we approximate 344
.
.
.
I
1.0 r
.
.
.
.
I
1.5
(fm)
Fig. 1. Solid lines: Monte Carlo (upper) and modified (lower) potentials. Dashed and dotted lines: the potentials of refs. [2] and [3], respectively, shifted to coincide with the modified potential for r = 0.5 fm.
and, from ref. [7]: Ac = 30.19Aw
.
0•5
fermionic contributions at the higher energies involved in quarkonia physics with their perturbative calculation. So we retain the form (2.5), (2.6) for the potential and the values (2.9) for A2w/O, a o and al, and take for f0, fl and A J A w the values (2.7), (2.8) for Nf = 3. The Monte Carlo potential and this modification are shown in fig. 1, together with two sample successful phenomenological potentials; other potentials are very similar to these in the range 0.1-1 fm (see ref. [1]). Given the string tension o, the potential is fully determined, except for an additive constant. This constant could in principle be determined by Monte Carlo calculations, e.g. by directly measuring bound state masses in the quenched approximation. Work is in progress in trying to extract this constant from Wilson loop values only.
3. Results• We use the phenomenological value ¢~- = 420 MeV, obtained from the Regge slope of
Volume 147B, number 4,5
8 November 1984
PHYSICS LETTERS
Table 1 ~k family for m c = 1.70 GeV; experimental data are taken from ref. [11]. Numbers in brackets are from experiment. State
Mass (GeV)
r(e + e - ) / r ( l S -, e + e-)a)
(t)2/e
1S 1P 2S 1D 2P 3S 2D 3P 4S 3D 5S
3.10 3.51 3.69 3.80 3.96 4.11 4.19 4.34 4.48 4.53 4.80
1.00
0.20 0.22 0.26 0.25 0.29 0.32 0.32 0.35 0.38 0.38 0.43
(3.10) (3.50) (3.69) (3.77)
0.48 (0.41 + 0.15)
(4.03 + 0.05) (4.16_+ 0.02)
0.34 ( - 0.2)
(4.42_+ 0.01)
0.26 0.22
2>
(( r 2 ))1/2 (fm)
0.40 0.62 0.79 0.81 0.97 1.12 1.13 1.27 1.40 1.41 1.65
a)F(1S --, e+e - ) = 6.17 keV (experiment: 4.6 _+ 1.4 keV).
t h e p t r a j e c t o r y . T h i s gives, u s i n g (2.9), t h e v a l u e
[71
(3.1)
A~ss = 50A w = 2 0 0 M e V
( c o m p a r e w i t h ref. [9]). A l l t h e r e s u l t s are fairly i n s e n s i t i v e to a 100 M e V u n c e r t a i n t y in t h e q u a r k m a s s e s ( c o m p e n s a t e d b y a n a p p r o p r i a t e shift in the a d d i t i v e c o n s t a n t ) , so w e c a n e s t i m a t e t h e m f r o m e x p e r i m e n t a l d a t a . W e t a k e for t h e c h a r m q u a r k m a s s
m c = m o - 150 M e V = 1700 M e V .
(3.2)
T h e a d d i t i v e c o n s t a n t a n d the b e a u t y q u a r k m a s s a r e t h e n o b t a i n e d b y fixing t h e m a s s e s of t h e g r o u n d s t a t e s ~k a n d 7' to t h e c o r r e c t e x p e r i m e n t a l v a l u e s . By n u m e r i c a l c a l c u l a t i o n , w e o b t a i n t h e r e s u l t s s h o w n i n t a b l e s 1 a n d 2; the l e p t o n i c w i d t h s a r e c a l c u l a t e d via [10]
F(nS---, e+e -) = { 16~e2qa2/[m(nS)l 2} × [1 - ( 1 6 / 3 ¢ r ) a s ( 2 m q )
] [ ~ , s ( 0 ) 12.
(3.3)
R e l a t e d u n c e r t a i n t i e s are d i s c u s s e d in ref. [1]. T h e m a x i m u m l e v e l shifts for a 100 M e V v a r i a t i o n in m q a r e 20 M e V f o r t h e ~k a n d 3 M e V f o r t h e T
Table 2 T family for m b = 5.07 GeV; experimental data are taken from refs. [12] and [13] (5S state). Numbers in brackets are from experiment. State
Mass (GeV)
r(e + e - ) / r ( l S --, e + e- )a)
( U2/C 2 )
( ( r 2 ))1/2 (fm)
1S 1P 2S 1D 2P 3S 2D 3P 4S 4P 5S
9.46 9.87 9.99 10.11 10.22 10.33 10.41 10.51 10.60 10.75 10.84
1.00
0.070 0.064 0.072 0.068 0.076 0.083 0.081 0.088 0.095 0.099 0.107
0.23 0.39 0.50 0.53 0.64 0.73 0.75 0.85 0.94 1.04 1.11
(9.46) (9.90) (10.02)
0.47
(0.42 _+0.02)
(10.26) (10.36)
0.35
(0.31 +_0.02)
(10.58)
0.29 (0.24 + 0.03)
(10.87)
0.26
a)F(1S --" e + e - ) = 1.17 keV (experiment: 1.22 +_0.03 keV). 345
Volume 147B, number 4,5
PHYSICS LETTERS
f a m i l i e s ; o t h e r q u a n t i t i e s are e v e n m o r e stable. A 10 M e V v a r i a t i o n i n x/b- gives a level shift u p to 30 M e V a n d a 5% v a r i a t i o n i n F ( e ÷ e - ) for the two families. T h e s e results c o m p a r e well with experim e n t a l d a t a (see ref. [11], a n d ref. [12] for the T f a m i l y ) : t h e m a s s differences are correct w i t h i n the e x p e c t e d - 5% relativistic c o r r e c t i o n s for the T a n d e v e n b e t t e r for the ~k families; T l e p t o n i c w i d t h s are also correct w i t h i n the e s t i m a t e d 10% c o r r e c t i o n s [1]. F i n a l l y , f r o m the b e a u t y q u a r k m a s s we o b t a i n rn B = m D - m c + m b = 5.24 G e V .
(3.4)
I w i s h to t h a n k C. R e b b i for p r o v i d i n g his r e s u l t s a n d for m a n y h e l p f u l discussions. T h i s w o r k was p a r t i a l l y s u p p o r t e d b y the D e l l a R i c c i a Foundation.
References [1] W. Buchmialler and S.-H.H. Tye, Phys. Rev. D24 (1981) 132, and references therein.
346
8 November 1984
[2] E. Eichten et al., Phys. Rev. D17 (1978) 3090; D21 (1980) 203. [3] G. Bhanot and S. Rudaz, Phys. Lett. 78B (1978) 119. [4] A. Martin, Phys. Lett. 93B (1980) 338. [5] M. Creutz, Proc. Fourth Johns Hopkins Workshop on Current problems in particle theory (Bonn, 1980); J.D. Stack, Phys. Rev. D27 (1983) 412; ITP-Santa Barbara preprint No. 136 (1983). [6] D. Barkai, K.J.M. Moriarty and C. Rebbi, Brookhaven preprint BNL-34923 (1984). [7] A. Billoire, Phys. Lett. 104B (1981) 472. [8] H. Hamber and G. Parisi, Phys. Lett. 47 (1981) 1792; Phys. Rev. D27 (1983) 208; E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795; F. Fucito et al., Nucl. Phys. B210[FS6] (1982) 407; D. Weingarten, Nucl. Phys. B215[FS7] (1983) 1; C. Bernard, T. Draper and K. Olnyk, Phys. Rev. D27 (1983) 227. [9] B. Adeva et al., Phys. Rev. Lett. 50 (1983) 2051. [10] R. Barbieri et al., Phys. Lett. 57B (1975) 455. [11] Particle Data Group, Phys. Lett. 111B (1983) 1. [12] P. Franzini, Columbia University preprint Nevis R # 1300 (1984). [13] P. Kass, talk XIXth Rencontre de Moriond, New particle production at high energies (March 1984).