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VOLUME 35, NUMBER 7 P H Y S I C A L R E V I E W LETTERS 18 AUGUST 1975
TABLE I. Upper limit on the cross sections for narrow resonances at three different masses.
2.25 GeV 3.1 GeV 3.7 GeV fc+tfv^
7 T ^ "
nrV PP K+K"
IT 7T~
PK' K+P pif TT+p
l x l O " 3 3
4 x l 0 " 3 3
. . . l x l O " 3 3
8X10**33
7X10" 3 3
2 X 1 0 ' 3 3
4X10" 3 2
2X10" 3 3
4 X 1 0 ' 3 5
8X10""35
4 X 1 0 ' 3 4
5X10"3 5
5X10"3 4
4X10" 3 4
4X10" 3 5
4X10" 3 3
4 x l 0 - 3 5
l x l O " 3 5
4X10" 3 5
2X10" 3 5
l x l O " 3 5
3X10" 3 5
3X10" 3 5
8X10" 3 6
5X10""34
7X10" 3 6
The spectrum increases with mass because of the increase of the acceptance, and then decreases again at higher mass because of the production mechanism. The individual spectra exhibit the expected statistical fluctuations, but there is no sharp peak in the mass region 1.25-5.0 GeV when all the overlapping spectra are compared.
Using a production mechanism for a particle in the c m . system of
d3o/dpi*2dp „ * = c exp(- 6p±*)/E*,
independent ofpu* and a persistent 5-standard-deviation peak above the background as a candidate for a new particle, we obtain typical upper limits of production times branching ratio for new resonances shown in Table I.
This result contradicts most of the present theoretical attempts to understand the existence of the J particle based on the charm model or the baryon-antibaryon model and so forth.5 It should be noted, however, that the analysis at the p res -
Recently Freund and Nambu1 have succeeded in constructing a model for the duality-violating <p -~pir and ^(3105) -» pir amplitudes which meets with quantitative success. Basically, the decays pro-
ent stage does not exclude an ordinary wide resonance of a width of a few hundred MeV. Finding such a resonance would depend on a detailed analysis of the acceptance and production mechanism, which we have not done.
We wish to thank the crew of the Brookhaven National Laboratory alternating-gradient synchrotron under Dr. H. Folsche and Dr. L. Smith for the superb operation of the accelerator. We thank also Ms. I. Schulz, Mr. J. Donahue, Mr. D. Osborne, Mr. C. Tourtellotte, and Mr. E. Weiner for technical assistance.
We wish to thank also Dr. S. L. Glashow, Dr. K. Johnson, Dr. F. Low, Dr. V. F. Weisskopf, and our colleagues at the Massachusetts Institute of Technology Center for Theoretical Physics for stimulating discussions. We also benefitted from discussions with T* D. Lee, B. W. Lee, M. Deutsch, R. R. Rau, S. Weinberg, T. T. Wu, and C. N. Yang.
* Permanent address: Institut de Physique Nucleaire, 91406 Orsay, France.
*J. J . Aubert et al., Phys. Rev. Lett 33, 1404, 1624 (1974), andNucl. Phys. B89, 1 (1975).
2J.-E. Augustin et al., Phys. Rev. Lett. j33, 1406 (1974).
3C. B a c c i ^ al., Phys. Rev. Lett. £3, 1408, 1649(E) (1974).
4J. J . Aubert et al., to be published. 5A. S. Goldhaber and M. Goldhaber, Phys. Rev. Lett.
34, 36 (1975); T. Applequist et al., Phys. Rev. Lett. 34, 365 (1975); E. E ich t en^ al., Phys. Rev. Lett. 34, 369 (1975); H. T. Nieh, T. T. Wu, and C. N. Yang, Phys. Rev. Lett. 34, 49 (1975); A. De Rujula and S. L. Glashow, Phys. Rev. Lett. 34, 46 (1975); M. K. Gail-lard, B. W. Lee, and J . L. Rosner, Rev. Mod. Phys. 47, 277 (1975).
ceed via pole dominance through an intermediary, the 0(JP= 1") meson, an SU(4) singlet and the daughter of the Pomeron. We extend the Freund-Nambu model to production and to other decays,
Duality Violation and the Production and Decay of New and Old Mesons*
John F. Bolzan,T Kevin A. Geer , Will iam F . P a l m e r , and Stephen S. Pinsky Department of Physics, The Ohio State University, Columbus, Ohio 43210
(Received 24 April 1975)
It is shown that the O meson, the Jp=l~ daughter of the Pomeron, is quite successful in predicting a number of rates for duality-violating processes.
419
VOLUME 35, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 18 AUGUST 1975
making analogous use of s eve ra l other mesons in the Pomeron family. Specifically, we consider a + b-ip + X, <p+X, ip'+X, 0(V)+X, and J)C + X, where a + b des ignates any hadronic init ial s ta te (e.g., ftp, pp, pd, ) which does not contain X or c quarks and in which a duality suppress ion mechan ism is r equ i red for s t rong production. We a l so calculate e+e~ - 0 ( 1 " ) -~ JX+M~«
The decay p r o c e s s e s we consider_are ip' -*/>ff, / ' - 7T7T, 0(2 +) - 7777, ip -+pp, and ip - AA.
The most famil iar pole-dominance model i s that of the p h o t o n - v e c t o r - m e s o n coupling with
fyv2/mv
p = 9:1:2:8 f o r p : o > : < ^ , (1)
where p depends on one ' s theory of SU(4) s y m m e t ry breaking. In the l i t e r a tu re p = 0, 2, 3, and 4 appear . With p = 3 a bes t fit i s found to V -~e *e ~.2
The pole couplings of p, co, (p, and $ to 0(1") and of 7], rj', and rjc to 0(0") a r e taken to be SU(4) s y m m e t r i c , with the nonet mixing scheme for the vec tor mesons and the Dashen et al.3 mixing scheme for the O" mesons . Thus
£ov=fov(S^ + <Pll + V)Om-), (2)
£ O P = / O P ( 0 . 5 3 T ] + 1 .65T? '+ 77 C )0 ( (T) , (3)
with SU(6) s u g g e s t i n g / o y « / O P . The t ensor (T) couplings of/, / ' , a n d / c , with nonet mixing, a r e
FIG. 1. Pole diagrams for duality-violating processes.
analogous to Eq. (2). With these definitions, a genera l inclusive
s t rong sp in-averaged production c r o s s sect ion for a duali ty-forbidden s ta te q r e f e r r i ng to Fig. 1(a) i s
S d2aq 1 A Qsu fa2
n dtdM2 2S(mQ2-m0
2)2 + i LfiL.
{m2 - m2)2 + mfT2
xEUuK2)! 1 + v> fon Kl^a2) ma2-m2+imirl
n*2foi A / * 0 m2-mn2 + im„r^
(4)
where Anj i s the exclusive ampli tude for producing one of a se t of o f f -mass - she l l mesons {n} plus a se t of pa r t i c l e s labeled by j . The se t of mesons n a r e the neutra l m e m b e r s of an SU(4) mult iplet with the s a m e Jp a s the O meson being considered, and which can be produced in duali ty-al lowed p r o c e s s e s . SU(4) s y m m e t r y for the off-mass ampli tude Anj(mq
2) impl ies Anj(m 2)/Au{m 2) = 1. Then
d2aq
dtdM2''
where
cPo^mJ) / , Oa_ / , dtdM*' ( m / - w 0
2 ) 2 + m 02 r (
fii_ ! - m 12 ) 2 + w 1
2 r i2 -F{m?), (5)
1 + £ - / f i l ! « t - m2 +imxT\
n*2JOi fo and d2a1(mq
2)/dtdMx2 i s the off-shell inclusive production c r o s s sect ion for the pa r t i c l e 1. Similar ly ,
for producing the O meson itself, r e f e r r i n g to Fig. 1(b), the express ion for O production i s obtained from Eq. (5) by s t r ik ing the f i r s t propagator t e r m and replac ing m2 by m0
2. As our f i r s t example let us calculate the production of the ^(3105) in a pro ton-pro ton coll ision and
i ts subsequent decay into e*e~ [Fig. 1(e)], where n is the o> and r i s the ip, as compared to a vec to r -dominated background [Fig. 1(f)].
' 2 ^ d2a ^ b g id3p + d3p"\d3p + d3p
:hl Zfo\ (P + +P")2 = m2 /• yu ?r,2(mJ-m0
2)2 Ity x $
«, (6)
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VOLUME 35, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 18 AUGUST 1975
where <ft=(l + / y p / / y J ' 2 , m w s W p , andVor = 0 - 1 3 7 GeV2 for m 0 = V2 GeV and 0.194 GeV2 for m0 = J3 GeV. Integrating this expression over the 20-MeV resolution of Aubert et al.,4 using the p = 3 parame-trization of the yF coupling,2 we find that Eq. (6) yields a signal-to-background ratio of 19 (105) for m0
2 = 2 (3) GeV2. These values are in rather good agreement with the Aubert et al.4 data which seem to show a back
ground of 2 or 3 events and a peak height of from 40 to 80 events. We should also keep in mind that our naive vector-dominated background is a lower-limit estimate.
With use of Eq. (5), the absolute cross section for ip production in terms of off-mass-shell u) production amplitude is
<Po» ^d2o"{m£) f 1.5XMT7 for mQ = /2 , , „ . dtdMx
2~ dtdM2 ) 8 . 0x l0" 7 for m 0 = V~3. {)
A check on the reliability of this estimate consists in using the same model to calculate the duality-forbidden (p production amplitude
dtp* d2o"{m*) (0.0042for m 0 = V2, ( )
dtdMx2~ dtM2 X (0.0040 for ra0 = V~3. ( }
In a recent experiment of Ayres et al.5 the total and differential cross sections for Ts"p-^cpn have been measured, the density matrix similarity with up — UM at higher energies noted, and an experimental ratio reported at 5 and 6 GeV/c,
ofj'p^vn) 0,0035 ± o.ooio, (9) o(n p-+wi) '
in good agreement with Eq. (8). The same approach yields the cross section for ^'(3700) production in terms of the ip cross section:
W ' / rfV y* / /0 , ,y42a"(m,^ (w) dtdM2\dtdM2) \f04, } dtdM2 \ dtdM2 / (0.18 for m0 = / 3 . V
Assuming that the off-shell co production amplitudes do not change appreciably between m / and m^2, we have, for the e +e" final state,
dtdM2
Using the lower limit for the width of the ip',G we find that (11) is >̂ 0.05(/o^//jfo^)2, where f^/f^ may be estimated if a common strong decay mode can be observed, e.g., the pti mode,
(12) T ^ p 7 r \mr2-m0
2J rn^-m2 mr3\f^) '
From Eq. (8) the cross section for 0(1") production is
ePa**1'-y^d2ow{m02) (0.019 for m 0 = Vr2,
dtdM 2~ dtdM2 X {0.013 for m 0 = / 3 . (13>
One may also ask whether the 0(1") might be found in the e*e" final state in the Aubert et al.4 experiment. Comparing the production process, Fig. 1(e), with n being the a; and r being the sum of the u>, <PJ and $, to a vector-meson-dominated background, Fig. 1(f), we find
d2aiO(l')-e + +e')f d2oh2 \"1"| (6.68xlO"4 for m0 = J2 GeV, d3p + d3p~ \d3p + d3p »o2 2.3xl0" 2 for m0=V~3 GeV. (14>
At the Stanford Linear Accelerator Center, the 0(1") effect in e*e~ -~0(1~) ^e+e" or /LL+JLL~ is also small, with little help from the interference effect because the 0(1") is so broad. A simple calculation yields
o<e+^ '*-O(l")- / i + iLL")/afe+^"-MV":QED)^10"6. (15)
Using Eq. (6) we can calculate the inclusive cross section for r]^ which will be mediated by the O(0")
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V O L U M E 35, N U M B E R 7 P H Y S I C A L R E V I E W L E T T E R S 18 AUGUST 1975
and the 77 and r\'\ we have, w i t h / 0 F ~ / 0 P and m^ <; 2 GeV,
" V ' j l x l O - ' (.6, dtdMx
2~dtdMx2'
Production of/' and / c may be analogously calculated, with the value oifOT inferred from decay data. The experimental limit Tf,-+2ir<12 MeV indicates that f0T<0.185 GeV (0.28 GeV) for raG
2 = 2 GeV2
(3 GeV2) implies r0 ( 2+}^2 7 r<69 MeV (30 MeV). Finally, the partial width for < (̂3105) -~pp may be calculated via the sequential decay chain ip-+0(l~)
-+u)-~pp, with the result
where gup-p2 is evaluated at the ip mass. Assuming ideal mixing and universal isospin current coupling,
we have
477 47T 4 4TT " '
on shell. Thus we have
r ^ ^ = £ s S ^ 2 4 ( 1 . 0 3 5 k e V ) for m 02 =2 GeV2. (18)
Similarly we have, from the sequential-pole model,1
- ^ = 0 . 0 1 1 5 ^ 4 T ^ 4 - <19>
Defining
gwpA™?) gwpp2(™J)' it is reasonable that x^l since we expect these amplitudes to extrapolate off shell in a qualitatively similar way. Thus we have
* A <p~+ pit
If we estimate further that
we have
fctffl«ijt^xj 1 ^ - ^ = 615 keV, (22) r -̂ r
r ^ - * 7 i > = " ^ ~ r ^ - > 3 7 T - ( 2 3 )
With x = l this is consistent with_experimental results.7
A similar calculation for the A A final state indicates r^-+AA/T0-^~O.9, reasonably close to the experimental result.7
*Work supported in pa r t by the U. S. Atomic Energy Commiss ion . tWork per formed in par t ia l fulfillment of the r e q u i r e m e n t s for the Ph . D. degree at The Ohio State Univers i ty ,
Columbus, Ohio 43210. 1 P . G. O. F reund and Y. Nambu, P h y s . Rev. Let t . 34, 1645 (1975). 2K. Gee r , G. B. Mainland, W. F . P a l m e r , and S. S. P insky, P h y s . Rev. D 11_, 2480 (1975); D. Yennie, P h y s . Rev.
Le t t . 34, 239 (1975). 3R. F . Dashen, I. J . Muzinich, B . W. Lee , and C. Quigg, F e r m i l a b Repor t No. FNAL 75/18-THY (unpublished). 4 J . J . Aubert et al„ P h y s . Rev. Let t . 3.3, 1404 (1974). 5D. S. A y r e s , R. Diebold, A. F . Greene , S. L. K r a m e r , J . S. Levine, A. J . Pawlicki , and A. B . Wicklund, P h y s .
Rev. Let t . 32, 1463 (1974). 6M. K. Gai l la rd , B . W. Lee , and J . L. Rosner , F e r m i l a b Repor t No. FNAL 75/14-THY (unpublished). 7G. Hansen, in Proceed ings of an International Conference on High Energy P h y s i c s , P a l e r m o , Italy, 2 3 - 2 8 June
1975 (to be published) .
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