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Four quark states and open charm production
Volume 83B, number 2
PHYSICS LETTERS
7 May 1979
FOUR QUARK STATES AND OPEN CHARM PRODUCTION ~ Y. AFEK, C. LEROY 1 and B. MARGOLIS Phystcs Department, McGtll Untverstty, Montreal, Canada Received 4 January 1979
We discuss and calculate the contribution of resonances to open charm production through two-gluon anmhflation m nucleon-nucleon interaction. Four-quark resonances, if they exist, make an important contribution because of their strong couphng to two gluons.
The calculation o f open charm production m p - p interaction has been much discussed recently [1 ] what with their being suggestions from experiment [2] as to the size of the cross section. The approach has normally involved the calculation o f the lowestorder QCD contributions and this means mainly twogluon annihilation, once the D r e l l - Y a n contribution falls off. Although the experimental results are vague at the moment they have led some theorists to speculate that at energies available at present [3] "boundstate" effects may invahdate the dominance of the lowest-order QCD results. We consider here the contribution to open charm production from two-gluon production o f heavy resonances which then decay into open charm through modes such as DD. We can see whether this mechanism is an important one when compared with lowest-order QCD and investigate the feasibility of studying the nature of the resonances coupling most strongly to two gluons. Our results indicate that two-quark states couple less strongly than certain proposed four-quark states. These four-quark states have the flavour content of c~q~, where q is a light quark. There are several discussions in the recent literature [ 4 - 7 ] concerning the possible existence o f such states and the possibility that the lowest o f these may correspond to the state Supported m part by the Natural Sciences and Engineering Research Councd of Canada and the Quebec Department of Education. 1 Also at the Laboratoire de Physique Nucl6alre, Umverslt6 de Montreal. 238
with mass 2.82 GeV [7] ,1 which we call X, resulting from a relatively feeble one-photon decay of the 4;(3.1). We show here that the production o f such a colour-smglet state in p - p interaction through twogluon annihilation is substantially larger than the lowest-order QCD contribution when the two gluons are in a colour-singlet state. The identification of the X(2.82) with c2q~ having isospin I = 0 (Sc) and I = 1 (8c) is connected with problems in the charmomum model. These problems are that (i) the branching ratio B(X -+ "/7) is higher than expected; (ii) the hyperfine splitting needed to reproduce the ( 3 . 1 ) - X mass difference is considerably larger than seems reasonable, and (ili) the magnetic dipole transition ff -+ X7 is b y about a factor 20 smaller than if X were the c-6 0 state, ~/cThe identification of X with one or b o t h of S c, 8 c has been suggested [5] in analogy with the mterpretaDirect estimates using the same picture for the light and heavy quarks yield [5] that the lowest c~q~ state is at about 3.6 GeV. This will enlarge the contribution of these states to open charm but will make our estimate of p(c-6q~ gg) less certain. Our attitude is 0) we accept the existence of four-quark states independently of the details of the calculations [6,7], 0i) we take the ideal mixing and the magnetic interaction from ref. [4]. These ingredients are enough for constructing the flavour and SU(3)colour × SU(2)spln parts of the wave function. A modificataon in the details of the bag potential might change the overall mass scale.
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PHYSICS LETTERS
tion o f S*(980) and 6(980) as ssqq states lying just below the 4~ meson. The larger width into 7")' o f a four-quark state than from a two-quark state is explained naturally by the sphtting o f the four-quark states into two vector quark states which then decay into a photon through vector dominance. The decay does not involve glue exchange or, m another language, OZI vaolation. This decay mode is proportional to the component in the fourquark wave functaon corresponding to VV (V = vector, colour singlet). Since the wave function also has a component proportional to VV (V - vector and colour octet), there is also a considerable decay width into two gluons. Nevertheless as we shall see below, B(X -+ 77) is expected to be relatively large for the four-quark states [7]. The strong coupling o f fourquark states to two ghions leads to a substantial contributlon o f these states to open charm production in p - p colhsions if there are such states with mass > 2M D. We start by discussing the open charm production in p - p collismns: When the process pp ~ c~ + anything is dominated b y a subprocess [1] ri--+ cF, where r is a quark or a gluon, then a ( p p -~ cF + anything) s
= r=q'q'g~ iMp)
r
1
(1)
d~s o(s', r-f~ c-d)7 rf ~ fP(x)fPr(r/x)'
7 May 1979
1S small. We need the two-ghion decay width of the states which in the non-relativistic quark model for Sstates is given by P(i -+ gg)
8 2 2 = ~(as/Mt )l R,(0)[2
where Ri(O ) is the radial wave function of the cF system at the origin. For higher-L states we get similar expressions [9]. We have used the potential
87r( 1 - r/ro~ l V(r) = ~ \i-+~O ! ~ )
+ Br + V0 ,
o(s',
r?- --" CC--),e s
= ~ 8 (s' - M2)(Srr2/M~)
o(gg -+ c~, s ' )
(2)
+ (1 -- ')')1/2'i
- (~- + r~ v)(1
- ,),)1/2
,
(5)
" " /A ' 2 " ) and 7 = 4Mc/s 2 ' , where A with a s = ~12n m, - 1(s = 5 0 0 - 7 0 0 MeV. Fig. 1 shows the contribution of the sum of resonances given by formula (2) to open charm production compared to the QCD lowest-order result, using the gluon distribution function fg(X) = ~-(1 -
P(i-+ rr-)(2 J / + 1).
(4)
w i t h B = 849 MeV 2, V0 = 464 MeV and r 0 = 0.221 fm to calculate the properties of the charmonium system inchidlng the width (3). The quark mass is M c = 1.1 GeV to fit the levels and leptonic widths of the cg system ,2 Results are given in table 1. Most o f the contnbutlon to the sum (2) comes from n < 5. The l = 2 states contribute less than 5%. For the situation where r, ?are not in a colour-slnglet state (there is a statistical factor 63" 1 in favour of this case) there are no resonance contributions to open charm production and the cross section is calculated here using lowest-order QCD diagrams. Here again the dominant contribution is from gluon annihilation and [10]
7"f~ 2
where r = s'/s, fP(x) is the quark (q) or gluon (g) momentum distributmn m the proton, s is the total energy squared in the C.M. of the pp system and 4MD2 is Smin for open charm production. In the case where r, g are emitted in such a way that they are in a colour-singlet state we can describe a contribution to eq. (1) from resonances which couple to two gluons [8],
(3)
x)5/x.
(6)
The overall features of the results are that"
1
Let us first consider the contribution o f states i from the c~ charmonium spectrum. The sum over r In eq. ( l ) is then dominated by the term r = g (ghions). The set of charmonium states which contribute substantially are n l S1, n3P0, n3P2, with n large enough so that M, > 2 M D. The contribution o f states with l = 2
+2 From potential (4) and using the values of the parameters indicated m the text, one finds M~ = 3.096 GeV, F($ -+ e+e-) = 4.68 keV, M$, = 3.71 GeV, F(~p' -+ e+e-) _- 2 37 keV and M T = 9.469 GeV, I'(T ~ e+e-) = 1.48 keV, M T' = 10.000 GeV, r(T' ~ e+e-) = 0.65 keV, the quark masses bemgM c = 1.100 GeV for the 7/s andM b = 4 530 GeV for the T's. 239
Volume 83B, n u m b e r 2
PHYSICS LETTERS
Table 1 The masses and the two-gluon decay widths of the S and P c h a r m o n i u m states are listed as obtained from potential (4)
S states
P states
n
Masses (GeV)
F( t ~ gg) (MeV)
0 1 2 3 4 5 6 7 8 9 10
3.096 3.710 4.170 4.558 4.912 5.239 5.545 5.836 6.114 6.381 6.638
12.69 6.44 4.55 3.57 2.94 2.51 2 19 1.93 1.74 1 57 1 44
0 1 2 3 4 5 6 7 8 9 10
3.513 3.995 4.402 4.766 5.101 5.414 5.710 5.992 6.262 6.523 6.775
2.11 1.88 1 58 1.35 1.16 1.01 0.89 0.79 0.70 0 63 0.57
(a) the general shape of the energy dependence of the open charm production cross section is similar for resonance production and lowest-order QCD, being dictated by the gluon distribution which favours low s' contributions, (b) the QCD results are much larger than the contribution of the resonances, mainly because of the 1/64 "statistmal" factor for the production of a colour singlet from the two gluons. We note here that introducing a momentum transfer dependence into the gluon distribution function given by eq. (5) produces only modest changes. This is so because of the dominance of the small s' or x region for which no large q2 dependence is expected. Let us now estimate the possible contribution of four-quark states. It has been suggested [6] that there could be a substantial spectrum of four-quark states of the type c~q~ in the mass region starting at the X(2.82). Let us estimate the two-gluon decay width of such a state. We first compute the ratio for the lowest I = 0 cgq~ state, Sc, P(S c -+ gg)/P(S c ~ 77), using a vector dominance type picture. We consider 240
7 May 1979
looI b
II
°
.f
/
/ 1 ~
/7
/ i
i ]
0
I 40
J
7F (GeV) I
80
I
I
120
I
I 160
I
I 200
Fig. 1. The band represents the contribution o f the four-quark states to the open charm production. The upper (lower) limit corresponds to I'(S~ ~ gg) = 410 MeV (r(S~ ~ gg) = 82 MeV) with a s = 0.3 (see text). The dashed curve is the QCD prediction and the d o t - d a s h e d curve represents the contribution o f c h a r m o n i u m states to the open charm production.
the decomposition of this state as [4,5] I Sc) = 0.328 IV1V2) - 0.393 IV1 • V 2) + 0.743 IP1P 2) - 0.432 ]P 1 • P2 ) ,
(7)
where bold face symbols denote a colour octet and V and P correspond to vector and pseudo-scalar meson states made from quark-antiquark. Our picture for the S c is V 1 = ff and V 2 = co. The two photons then couple one each to V 1 and V 2 and the ghions to V 1 and V 2. We have then that F(Sc -+gg)
(-0.393~ 2 ~s2 ( 2_3/2 f
P(Sc--~ T'/i = \ 0 . - ~ ]
c~2
~ \ - 5 ~ " 5 ] = 3.63 ~ .
(8)
The similar ratio for the c~ state r~c is [9] P(rlc ~ g g )
9 ~2
I'(rlc ~ 77) - 8 a2
(9)
We have then that the two-gluon width in ratio to the two-photon width is about three times as big for the four-quark state as for the two-quark states. Further one expects the partial width for decay into two-pho-
Volume 83B, number 2
PHYSICS LETTERS
tons to be bigger for S c than for r/c. This is one o f the reasons for identifying the X(2.82) with the fourquark states Sc, 8 c. Using data from the one experiment which has been done on the reaction [11 ]" rr
+ p-+73` + n ,
(10)
with 34(77) = (2.88 -+ 0.06) GeV, it has been deduced that F(S c -+ 77) ~ 10 P(r/c -* 77). The value of V(r/c -+ 3'7) obtained using the charmonium model described b y the potential of eq. (4) is 6.68 keV. This leads to P(S c -+ gg) ~ 410 MeV using eq. (8) with as = 0.3. This is to be compared with F(r/c-+ gg) = 12 7 MeV. The verification o f this estimate for Y(S c -~ gg) awaits further experimental data on reaction (10). We can make another estimate for F(S c -+ gg) with the help of the argument o f Holmgren and Pennlngton [12] along with vector dominance and using eqs. (8) and (9). The authors of ref. [12] argue that the decay width o f a four-quark state into a pair o f twoquark mesons should be larger than the similar twoquark meson decay width b y a factor 0r/c~0s)2, where c~0 is the g l u o n - q u a r k couphng constant squared, corresponding to soft gluon emission. Using this result and vector dominance we then write I'(Sc -+ 3'3')/P(r/c -+ 3'3') ~ (lr/c~°s)2"
(11)
Here we assume that the hadronic states under consideration decompose into co~ which each emits a ~hoton. The authors o f ref. [12] estimate that (rr/as) 2 = 2. Combining eqs. (8), (9) and (11), we have then that P(S c -+ gg) ~ 82 MeV We have then two esUmates for P(S c --* gg), consistent to within a factor of five to be compared with lP(r/c -+ gg) which we find to be 12.7 MeV using eq. (3) with c~s = 0.3. This indicates a strong coupling o f two gluons to four-quark states. Given then a spectrum of four-quark states of similar density to that for two-quark states, we estimate a contribution to open charm production 6 - 3 0 times as great as from charmonlum states. Since the purpose o f this artxcle ~s only to provide estimates, we take the density o f the four-quark states equal to that of the charmonium states as a rough approximation. This leads to a contribution shown by the band in fig. 1. More detailed calculations clearly depend on a full study o f the spectrum o f the four-quark states m the mass region in question. It is to be expected that in general two gluons will couple relatively strongly to
7 May 1979
all these four-quark states. We conclude then that the four-quark states have a substantial influence on open charm production in p - p collisions. From fig. I, one sees a contribution similar in size to the QCD lowest-order g h i o n - g l u o n nonsinglet annihilation contributions and much larger than the colour-singlet QCD contribution (using a colour factor that assumes the two gluons constitute a colour sanglet). It could be of interest to search for these states through typical decay modes such as 7")' m p - p or ~ - p interactions We note m this connection that the relatively large cross section for production of fourquark states when compared with the cross section for production of the related two-quark states, e.g., S e production versus r/c production, is also a feature of peripheral production presumably mediated by Regge exchanges [7]. After completion of the manuscript, we received a pceprint [13] where the contribution o f charmomum bound-states to open charm production was also estimated. These authors differ from our estimate mainly by their consideration of the bound states below DD but above c~ thresholds. However, they do not discuss four-quark states and their possible contribution to open charm production. We would like to thank N. de Takacsy for providing us with his program to calculate the potential used m this paper and for helpful discussions.
References [1] See for instance L Jones and H. Wyld, Phys. Rev D17 (1978) 1782; J Babcock, D. Slvers and S. Wolfram, Phys Rev 18 (1978) 162, B.L Combridge, CERN preprlnt TH-2574 (October 1978). [2] A G Clark et al~, Phys Lett 77B (1978) 339, L. Baum et al, Phys Lett. 77B (1978) 337, P. Ahbran et al., Phys. Lett 74B (1978) 134, T. Hansl et al, Phys. Lett. 74B (1978) 139, P.C. Bosettl et al., Phys Lett 74B (1978) 143, D. Spelbring et al., Phys. Rev. Lett. 40 (1978) 605, R Lipton et al., Phys. Rev Lett 40 (1978) 608 [3] H. Fritzsch and K H Streng, Phys Lett. 77B (1978) 299 [4] R.L. Jaffe, Phys. Rev. Dl5 (1977) 267,281 [5] A De Rujula and R.L Jaffe, Proc Conf. Experimental meson spectroscopy (Northeastern University, Boston, 1977). 241
Volume 83B, number 2
PHYSICS LETTERS
[6] H J Llpkin, N. Isgur and H R. Rubmstein, Phys. Lett. 78B (1978) 295 [7] G Ellam, B Margohs and S Rudaz, Phys Lett 80B (1979) 306. [8] C.E. Carlson and R Suaya, Phys. Rev DI8 (1978) 760 [9] V A. Novlkov et al, Phys Rep 41C (1978) 1
242
7 May 1979
[10] See for instance M Gltick, J.F. Owens and E. Reya, Phys Rev D17 (1978) 2324 [11] W D. Apel et al., Phys Lett 72B (1978) 500 [12] S O Holmgren and M R Pennmgton, Phys Lett. 77B (1978) 304. [13] C.E. Carlson and R. Suaya, SLAC preprint SLAC-PUB2212 (October 1978).
PHYSICS LETTERS
7 May 1979
FOUR QUARK STATES AND OPEN CHARM PRODUCTION ~ Y. AFEK, C. LEROY 1 and B. MARGOLIS Phystcs Department, McGtll Untverstty, Montreal, Canada Received 4 January 1979
We discuss and calculate the contribution of resonances to open charm production through two-gluon anmhflation m nucleon-nucleon interaction. Four-quark resonances, if they exist, make an important contribution because of their strong couphng to two gluons.
The calculation o f open charm production m p - p interaction has been much discussed recently [1 ] what with their being suggestions from experiment [2] as to the size of the cross section. The approach has normally involved the calculation o f the lowestorder QCD contributions and this means mainly twogluon annihilation, once the D r e l l - Y a n contribution falls off. Although the experimental results are vague at the moment they have led some theorists to speculate that at energies available at present [3] "boundstate" effects may invahdate the dominance of the lowest-order QCD results. We consider here the contribution to open charm production from two-gluon production o f heavy resonances which then decay into open charm through modes such as DD. We can see whether this mechanism is an important one when compared with lowest-order QCD and investigate the feasibility of studying the nature of the resonances coupling most strongly to two gluons. Our results indicate that two-quark states couple less strongly than certain proposed four-quark states. These four-quark states have the flavour content of c~q~, where q is a light quark. There are several discussions in the recent literature [ 4 - 7 ] concerning the possible existence o f such states and the possibility that the lowest o f these may correspond to the state Supported m part by the Natural Sciences and Engineering Research Councd of Canada and the Quebec Department of Education. 1 Also at the Laboratoire de Physique Nucl6alre, Umverslt6 de Montreal. 238
with mass 2.82 GeV [7] ,1 which we call X, resulting from a relatively feeble one-photon decay of the 4;(3.1). We show here that the production o f such a colour-smglet state in p - p interaction through twogluon annihilation is substantially larger than the lowest-order QCD contribution when the two gluons are in a colour-singlet state. The identification of the X(2.82) with c2q~ having isospin I = 0 (Sc) and I = 1 (8c) is connected with problems in the charmomum model. These problems are that (i) the branching ratio B(X -+ "/7) is higher than expected; (ii) the hyperfine splitting needed to reproduce the ( 3 . 1 ) - X mass difference is considerably larger than seems reasonable, and (ili) the magnetic dipole transition ff -+ X7 is b y about a factor 20 smaller than if X were the c-6 0 state, ~/cThe identification of X with one or b o t h of S c, 8 c has been suggested [5] in analogy with the mterpretaDirect estimates using the same picture for the light and heavy quarks yield [5] that the lowest c~q~ state is at about 3.6 GeV. This will enlarge the contribution of these states to open charm but will make our estimate of p(c-6q~ gg) less certain. Our attitude is 0) we accept the existence of four-quark states independently of the details of the calculations [6,7], 0i) we take the ideal mixing and the magnetic interaction from ref. [4]. These ingredients are enough for constructing the flavour and SU(3)colour × SU(2)spln parts of the wave function. A modificataon in the details of the bag potential might change the overall mass scale.
Volume 83B, number 2
PHYSICS LETTERS
tion o f S*(980) and 6(980) as ssqq states lying just below the 4~ meson. The larger width into 7")' o f a four-quark state than from a two-quark state is explained naturally by the sphtting o f the four-quark states into two vector quark states which then decay into a photon through vector dominance. The decay does not involve glue exchange or, m another language, OZI vaolation. This decay mode is proportional to the component in the fourquark wave functaon corresponding to VV (V = vector, colour singlet). Since the wave function also has a component proportional to VV (V - vector and colour octet), there is also a considerable decay width into two gluons. Nevertheless as we shall see below, B(X -+ 77) is expected to be relatively large for the four-quark states [7]. The strong coupling o f fourquark states to two ghions leads to a substantial contributlon o f these states to open charm production in p - p colhsions if there are such states with mass > 2M D. We start by discussing the open charm production in p - p collismns: When the process pp ~ c~ + anything is dominated b y a subprocess [1] ri--+ cF, where r is a quark or a gluon, then a ( p p -~ cF + anything) s
= r=q'q'g~ iMp)
r
1
(1)
d~s o(s', r-f~ c-d)7 rf ~ fP(x)fPr(r/x)'
7 May 1979
1S small. We need the two-ghion decay width of the states which in the non-relativistic quark model for Sstates is given by P(i -+ gg)
8 2 2 = ~(as/Mt )l R,(0)[2
where Ri(O ) is the radial wave function of the cF system at the origin. For higher-L states we get similar expressions [9]. We have used the potential
87r( 1 - r/ro~ l V(r) = ~ \i-+~O ! ~ )
+ Br + V0 ,
o(s',
r?- --" CC--),e s
= ~ 8 (s' - M2)(Srr2/M~)
o(gg -+ c~, s ' )
(2)
+ (1 -- ')')1/2'i
- (~- + r~ v)(1
- ,),)1/2
,
(5)
" " /A ' 2 " ) and 7 = 4Mc/s 2 ' , where A with a s = ~12n m, - 1(s = 5 0 0 - 7 0 0 MeV. Fig. 1 shows the contribution of the sum of resonances given by formula (2) to open charm production compared to the QCD lowest-order result, using the gluon distribution function fg(X) = ~-(1 -
P(i-+ rr-)(2 J / + 1).
(4)
w i t h B = 849 MeV 2, V0 = 464 MeV and r 0 = 0.221 fm to calculate the properties of the charmonium system inchidlng the width (3). The quark mass is M c = 1.1 GeV to fit the levels and leptonic widths of the cg system ,2 Results are given in table 1. Most o f the contnbutlon to the sum (2) comes from n < 5. The l = 2 states contribute less than 5%. For the situation where r, ?are not in a colour-slnglet state (there is a statistical factor 63" 1 in favour of this case) there are no resonance contributions to open charm production and the cross section is calculated here using lowest-order QCD diagrams. Here again the dominant contribution is from gluon annihilation and [10]
7"f~ 2
where r = s'/s, fP(x) is the quark (q) or gluon (g) momentum distributmn m the proton, s is the total energy squared in the C.M. of the pp system and 4MD2 is Smin for open charm production. In the case where r, g are emitted in such a way that they are in a colour-singlet state we can describe a contribution to eq. (1) from resonances which couple to two gluons [8],
(3)
x)5/x.
(6)
The overall features of the results are that"
1
Let us first consider the contribution o f states i from the c~ charmonium spectrum. The sum over r In eq. ( l ) is then dominated by the term r = g (ghions). The set of charmonium states which contribute substantially are n l S1, n3P0, n3P2, with n large enough so that M, > 2 M D. The contribution o f states with l = 2
+2 From potential (4) and using the values of the parameters indicated m the text, one finds M~ = 3.096 GeV, F($ -+ e+e-) = 4.68 keV, M$, = 3.71 GeV, F(~p' -+ e+e-) _- 2 37 keV and M T = 9.469 GeV, I'(T ~ e+e-) = 1.48 keV, M T' = 10.000 GeV, r(T' ~ e+e-) = 0.65 keV, the quark masses bemgM c = 1.100 GeV for the 7/s andM b = 4 530 GeV for the T's. 239
Volume 83B, n u m b e r 2
PHYSICS LETTERS
Table 1 The masses and the two-gluon decay widths of the S and P c h a r m o n i u m states are listed as obtained from potential (4)
S states
P states
n
Masses (GeV)
F( t ~ gg) (MeV)
0 1 2 3 4 5 6 7 8 9 10
3.096 3.710 4.170 4.558 4.912 5.239 5.545 5.836 6.114 6.381 6.638
12.69 6.44 4.55 3.57 2.94 2.51 2 19 1.93 1.74 1 57 1 44
0 1 2 3 4 5 6 7 8 9 10
3.513 3.995 4.402 4.766 5.101 5.414 5.710 5.992 6.262 6.523 6.775
2.11 1.88 1 58 1.35 1.16 1.01 0.89 0.79 0.70 0 63 0.57
(a) the general shape of the energy dependence of the open charm production cross section is similar for resonance production and lowest-order QCD, being dictated by the gluon distribution which favours low s' contributions, (b) the QCD results are much larger than the contribution of the resonances, mainly because of the 1/64 "statistmal" factor for the production of a colour singlet from the two gluons. We note here that introducing a momentum transfer dependence into the gluon distribution function given by eq. (5) produces only modest changes. This is so because of the dominance of the small s' or x region for which no large q2 dependence is expected. Let us now estimate the possible contribution of four-quark states. It has been suggested [6] that there could be a substantial spectrum of four-quark states of the type c~q~ in the mass region starting at the X(2.82). Let us estimate the two-gluon decay width of such a state. We first compute the ratio for the lowest I = 0 cgq~ state, Sc, P(S c -+ gg)/P(S c ~ 77), using a vector dominance type picture. We consider 240
7 May 1979
looI b
II
°
.f
/
/ 1 ~
/7
/ i
i ]
0
I 40
J
7F (GeV) I
80
I
I
120
I
I 160
I
I 200
Fig. 1. The band represents the contribution o f the four-quark states to the open charm production. The upper (lower) limit corresponds to I'(S~ ~ gg) = 410 MeV (r(S~ ~ gg) = 82 MeV) with a s = 0.3 (see text). The dashed curve is the QCD prediction and the d o t - d a s h e d curve represents the contribution o f c h a r m o n i u m states to the open charm production.
the decomposition of this state as [4,5] I Sc) = 0.328 IV1V2) - 0.393 IV1 • V 2) + 0.743 IP1P 2) - 0.432 ]P 1 • P2 ) ,
(7)
where bold face symbols denote a colour octet and V and P correspond to vector and pseudo-scalar meson states made from quark-antiquark. Our picture for the S c is V 1 = ff and V 2 = co. The two photons then couple one each to V 1 and V 2 and the ghions to V 1 and V 2. We have then that F(Sc -+gg)
(-0.393~ 2 ~s2 ( 2_3/2 f
P(Sc--~ T'/i = \ 0 . - ~ ]
c~2
~ \ - 5 ~ " 5 ] = 3.63 ~ .
(8)
The similar ratio for the c~ state r~c is [9] P(rlc ~ g g )
9 ~2
I'(rlc ~ 77) - 8 a2
(9)
We have then that the two-gluon width in ratio to the two-photon width is about three times as big for the four-quark state as for the two-quark states. Further one expects the partial width for decay into two-pho-
Volume 83B, number 2
PHYSICS LETTERS
tons to be bigger for S c than for r/c. This is one o f the reasons for identifying the X(2.82) with the fourquark states Sc, 8 c. Using data from the one experiment which has been done on the reaction [11 ]" rr
+ p-+73` + n ,
(10)
with 34(77) = (2.88 -+ 0.06) GeV, it has been deduced that F(S c -+ 77) ~ 10 P(r/c -* 77). The value of V(r/c -+ 3'7) obtained using the charmonium model described b y the potential of eq. (4) is 6.68 keV. This leads to P(S c -+ gg) ~ 410 MeV using eq. (8) with as = 0.3. This is to be compared with F(r/c-+ gg) = 12 7 MeV. The verification o f this estimate for Y(S c -~ gg) awaits further experimental data on reaction (10). We can make another estimate for F(S c -+ gg) with the help of the argument o f Holmgren and Pennlngton [12] along with vector dominance and using eqs. (8) and (9). The authors of ref. [12] argue that the decay width o f a four-quark state into a pair o f twoquark mesons should be larger than the similar twoquark meson decay width b y a factor 0r/c~0s)2, where c~0 is the g l u o n - q u a r k couphng constant squared, corresponding to soft gluon emission. Using this result and vector dominance we then write I'(Sc -+ 3'3')/P(r/c -+ 3'3') ~ (lr/c~°s)2"
(11)
Here we assume that the hadronic states under consideration decompose into co~ which each emits a ~hoton. The authors o f ref. [12] estimate that (rr/as) 2 = 2. Combining eqs. (8), (9) and (11), we have then that P(S c -+ gg) ~ 82 MeV We have then two esUmates for P(S c --* gg), consistent to within a factor of five to be compared with lP(r/c -+ gg) which we find to be 12.7 MeV using eq. (3) with c~s = 0.3. This indicates a strong coupling o f two gluons to four-quark states. Given then a spectrum of four-quark states of similar density to that for two-quark states, we estimate a contribution to open charm production 6 - 3 0 times as great as from charmonlum states. Since the purpose o f this artxcle ~s only to provide estimates, we take the density o f the four-quark states equal to that of the charmonium states as a rough approximation. This leads to a contribution shown by the band in fig. 1. More detailed calculations clearly depend on a full study o f the spectrum o f the four-quark states m the mass region in question. It is to be expected that in general two gluons will couple relatively strongly to
7 May 1979
all these four-quark states. We conclude then that the four-quark states have a substantial influence on open charm production in p - p collisions. From fig. I, one sees a contribution similar in size to the QCD lowest-order g h i o n - g l u o n nonsinglet annihilation contributions and much larger than the colour-singlet QCD contribution (using a colour factor that assumes the two gluons constitute a colour sanglet). It could be of interest to search for these states through typical decay modes such as 7")' m p - p or ~ - p interactions We note m this connection that the relatively large cross section for production of fourquark states when compared with the cross section for production of the related two-quark states, e.g., S e production versus r/c production, is also a feature of peripheral production presumably mediated by Regge exchanges [7]. After completion of the manuscript, we received a pceprint [13] where the contribution o f charmomum bound-states to open charm production was also estimated. These authors differ from our estimate mainly by their consideration of the bound states below DD but above c~ thresholds. However, they do not discuss four-quark states and their possible contribution to open charm production. We would like to thank N. de Takacsy for providing us with his program to calculate the potential used m this paper and for helpful discussions.
References [1] See for instance L Jones and H. Wyld, Phys. Rev D17 (1978) 1782; J Babcock, D. Slvers and S. Wolfram, Phys Rev 18 (1978) 162, B.L Combridge, CERN preprlnt TH-2574 (October 1978). [2] A G Clark et al~, Phys Lett 77B (1978) 339, L. Baum et al, Phys Lett. 77B (1978) 337, P. Ahbran et al., Phys. Lett 74B (1978) 134, T. Hansl et al, Phys. Lett. 74B (1978) 139, P.C. Bosettl et al., Phys Lett 74B (1978) 143, D. Spelbring et al., Phys. Rev. Lett. 40 (1978) 605, R Lipton et al., Phys. Rev Lett 40 (1978) 608 [3] H. Fritzsch and K H Streng, Phys Lett. 77B (1978) 299 [4] R.L. Jaffe, Phys. Rev. Dl5 (1977) 267,281 [5] A De Rujula and R.L Jaffe, Proc Conf. Experimental meson spectroscopy (Northeastern University, Boston, 1977). 241
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PHYSICS LETTERS
[6] H J Llpkin, N. Isgur and H R. Rubmstein, Phys. Lett. 78B (1978) 295 [7] G Ellam, B Margohs and S Rudaz, Phys Lett 80B (1979) 306. [8] C.E. Carlson and R Suaya, Phys. Rev DI8 (1978) 760 [9] V A. Novlkov et al, Phys Rep 41C (1978) 1
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[10] See for instance M Gltick, J.F. Owens and E. Reya, Phys Rev D17 (1978) 2324 [11] W D. Apel et al., Phys Lett 72B (1978) 500 [12] S O Holmgren and M R Pennmgton, Phys Lett. 77B (1978) 304. [13] C.E. Carlson and R. Suaya, SLAC preprint SLAC-PUB2212 (October 1978).