Nuclear Physics A~l(l984)593-402 @ North-Holland Publishing Company
NUCLEON-ANTINUCLEON BOUND STATES IN A QUARK REARRAN~~ME~ MODEL J.A. NISRANEN* Research
and A.M. GREEN
Institutefor Theoretical Physics, University of Helsinki, Siltavuorenpenger 20 C, SF-001 70 Helsinki 17, Finland
Received 21 May 1984 (Revised 21 June 1984) Abstract: The possibility of deeply-bound NN states is studied in a quark rearrangement plus a meson-exchange potential model. It is shown that very narrow (I’= I-10 MeV) may be possible even for L = 0 with binding energies varying up to EB = 800-900 proposed scheme. EB is, however, very dependent on the details of the short-range the meson-exchange part. A reasonable agreement with the present controversial obtained.
annihiIation bound states MeV in the treatment of data can be
1. Introduction The question of the existence of narrow NN bound nuclear states or of massive mesons (baryonium), only weakly coupled to “ordinary” mesonic decay channels, has been very intriguing not only theoretically but also experimentally. Both the NN bound states ‘*2) and baryonium ‘) have been given some theoretical justification in potential and quark models, respectively. In the former case-the one to be discussed in this paper - the reasonably “well-known” meson-exchange NN potential, obtained by the G-parity transfo~ation from the corresponding NN potential, can support several deeply-bound states in the absence of the annihilation potential ‘). However, when an annihilation interaction-fitted to the NN cross sections above threshold - is included, the resulting states will have huge widths, up to GeV’s [ref. ‘)I. One possible way to decrease the widths is to consider a part of the inelasticity as being simply due to pion-emitting transitions into these bound states. The resulting modest decrease (- 10%) in the true annihilation cross section can result in a reduction of the actual annihilation potential -and consequently the widths - by one order of magnitude ‘). However, the width of, for example, a ‘3P2-‘3Fz state bound by 180 MeV is still 150 MeV. On the other hand, experimentally 4-5 states are claimed to be seen in the y ray spectra of stopped p’s [refs. ‘*6)] with binding energies ranging from 200 MeV to 500 MeV and width estimates less than 20-30 MeV - sometimes even the upper limit being -6 MeV! But, it should be added that the existence and interpretation of these y-rays is a very controversial subject. * On leave of absence from the Department of Theoretical Physics, University of Helsinki, Helsinki, Finland. 593
594
a Fig. 1. The quark
b
rearrangement model for Nfl annihilation: (a) to three mesons; with one qij pair annihilating into the vacuum,
(b) to two mesons
2. The NR irrteraction
In the present paper the quark rearrangement mudel for Ns annihilation, developed in refs. 7-9) [see also the review in ref. ‘“)1, will be used to study the possibility of narrow bound states in the N%J system. This model, in its present stage of sophistication, considers the rearrangement of the three quarks and antiquarks into three s-wave mesons or into one p-wave meson (““PO could be the vacuum) plus two s-wave mesons (see fig, la) leading to a separable annihilation potential that acts only in the relative S or P N@-states, respectively, The model is able to explain up to 70% of the observed annihilation in NR scattering, so that from ref. “) one would presume the resulting widths of the bound NR states to be of the order of = 100 MeV. However, due to different meson thresholds 19), only part of the channels that are open in the scattering region are also active far below the Ns threshold. In the estimates of ref. ‘), for EC,,. < 400 MeV, an average energy dependence of the imaginary part of the annihilation potential is exponential being proportional to exp (aE,.,.) with u = 3-4 GeV- ‘. Therefore, at -200-500 MeV below the Ns threshold, one would expect a reduction in the annihilation rate by a factor =3-10. In refs. 8*9)it was also shown that a substantial amount of annihilation occurs through the chain of reactions NhNii*Afi
or
Ad+
MLM2M3a
By energy arguments one might expect that the highly excited states including isobars would be suppressed far below threshold, so effectively decreasing further the widths of the states. Furthermore, since the annihilation potential contains also a strongly attractive real part (even stronger than the imaginary part) the annihilation process, in spite of its effective repulsion from the imaginary part, cannot push the states out of the well. In fact, making the annihilation potential stronger increases the binding energy. The NN annihilation potential from refs. 8T9)is of the form
J. A. Niskanen,
595
A.M. Green / NN bound states
for L=O, 1. Here p is the size parameter of the mesons, so that the meson wave functions are, as a function of quark coordinates r, and r,,
rCl(md= NY=, XxJ+I
- r21Lg exp [-$P(h - F2,)‘l.
(2.2)
This /3 is related to the proton charge rms radius b by /3 = J& 2bA2. Here b is taken as 0.6 fm, the remaining 0.2 fm being attributed to the pion cloud. The complex energy-dependent factor I( TS, E) is a combination of three effects: (a) the phase-space integrals for the various final meson states (including the widths of the mesons T(m2), first considered as constants); (b) the SU(4) (for the n and 7’ SU(6)) overlap Clebsch-Gordan coefficients together with the angular momentum couplings necessary for the p-wave mesons; (c) an additional factor to account for the possibility that the pion is not a pure 1s qq state (taken here to be $ for each pion). Finally A -the overall strength - is the main free parameter of the model. In ref. ‘) two maxima - each exhausting about 50-70% of the annihilation at E (lab) = 100 MeV- were found for A = 15-20 MeV 9fm and h = 80-90 MeV * fm. Of these the lower value is preferred by qualitative theoretical arguments “) and because it also seems to give better S-wave branching ratios 18). Finally, the meson-exchange potential employed is a simplified version of the G-parity transformed Paris NN potential as used e.g. by Dover and Richard ‘I). On this is imposed a cut-off factor, F=(r/r,)“/(l
+(r/rJ”)
(2.3)
with r, = 0.6 fm, so that the meson exchanges are suppressed in the region where direct quark interactions, such as those generated by annihilation, are expected to be important. For finding bound-state solutions the numerical procedure was to iterate the homogeneous integral equation
1
Vann(r’, r”)$(r”)r”* dr” rr2 dr’.
+(r) =
(2.4)
Only the ground states were searched for. The results were checked to go smoothly to the real case bound ground state with no nodes, when the annihilation potential was switched off (A + 0).
3. Results Without any annihilation (A = 0) seven bound states were found for the above model, three S-wave, three P-wave and one D-wave state as shown in figs. 2 and 3. It is interesting to note that the tensor coupling in the T = 0 states is so strong that the probabilities for the L = J f 1 tensor coupled states are about equal - indicating that the tensor force completely dominates the dynamics in these cases. The
596
LA. Niskanen,
A.M. Green / NN bound states
Fig. 2. The trajectories of the bound state energies in the complex energy plane as a function of the strength A (in MeV . fm). The crosses are for the constant meson widths and the hollow circles for the meson widths with a realistic energy dependence eq. (3.1). The squares for the 3’S0 state are obtained with the meson exchange potential switched off; half of the imaginary part is shown to fit in scale. The horizontal axes are the real parts and the vertical the imaginary parts of the energies in MeV.
calculated S-D and P-F state percentages for A = 0 and A = 20 MeV . fm are, respectively, ‘3S1-‘3D,:
38.7-61.3% ,
43.9-5&l%
33S1-33DI:
94.3-5.7% ,
96.5-3.5% ;
‘3P2”3F$
46.2~53.8% ,
51.5-48.5% ;
r3D3-‘3Gs:
50.9-49.1% ,
(independent
;
of A) .
The reason why the L = J -k 1 component in the T = 0 states can be dominant is that the very strong positive tensor force has there a much larger attractive diagonal component than in the L = J - 1 case.
J.A. Niskanen, A.M. Green / NR bound states 10
20
30
597
40 1800
-100
1700
-200
1600
-300
1500
-400
mass
MeV
-900 -
-1000
40 L MeV.fm
Fig. 3. The real parts of the bound-state energies and the masses of the bound NR systems as functions of A. For the state ‘3D,-‘3G3 no annihilation is introduced in the model and so it is independent of ht.
In fig. 2 trajectories when the annihilation
of the complex total energy of the N6I systems are shown, potential is switched on and its strength A is varied over the
region considered most realistic in refs. 899). A general feature of the three-meson annihilation potential is a strongly attractive real part - in fact, much larger than the imaginary’ part. This disparity is enhanced in the deepest bound states where the imaginary potential is further reduced through the mesonic channels effectively closing as the energy decreases. A reflection of this is the linearly increasing binding energy, when A is increased, as shown in fig. 3. Two more weakly bound states, “So and 3’P,, are found when A = 10 MeV . fm. For small values of A also the imaginary part behaves rather linearly, with decreasing slope for large A’s in the case of the more weakly bound states (E “only” = -200 MeV). This deviation from the behaviour
of the real part is probably due to the stronger energy dependence of values of A, all widths, defined as twice the imaginary part Im V,,,. For reasonable of the energy, are remarkably small - especially in the deepest states. In fact, for the fuvoured ‘) value A = 15-20 MeV the widths are -50 MeV or less. This result is obtained when constant intrinsic meson widths at the resonance energy (meson mass) are used. When a realistic energy dependence is inserted into the meson widths - due to form-factor and phase-space effects -the widths of the NR bound states are further reduced, as seen later. Footnote added in proof: The state shown as 33P,, should be 33P1 as in fig. 2.
J.A. Niskanen, A.M. Green / NN bound states
598
TABLE Experimental
data versus the present
Experimental mass [Meal
I
predictions
Theoretical
Seen in experiment 1976 “)
1979 b)
1980 “)
with A = 10 and 20 MeV . fm
‘)
candidates and predicted masses [MeV]
d,
A = 10 MeV . fm
X
1828*2 1771* 1
X
X
X
1694*2
X
X
X
1638*3 1557*3 1507*4 1421*6 1383+6 1 1210*5
X
X
X X
X
X
X
x
-
-
“0
X
1711’
-
X X
X
?
-
X
?
-
-
X
x
-
?
?
13D
3
_13G
3
1838 1822 1773 1713 1693 1657 1568 1506 > 1377 1200
A=20 MeV . fm 1847 1742 1761 1702 1632 1626 1657 see the text
1346 1175
The quantum number assignments given are adequate with A = 10 MeV. fm, whereas for A = 20 MeV . fm they should be shifted. Crosses mean a confidence level 2 1.7~, while the question mark means that the state is seen with a statistical significance less than this. No sign means that the energy region was not covered in the experiment. Because of the model dependence and experimental uncertainties the state assignments-as well as the agreement between theory and experiment-should be considered only tentative. ‘) Ref. “). d, Ref. “). “) Ref. sa). ‘) Ref. 5b).
In table theoretical
1 the experimental data of refs. 5*6) are summarized together with the candidates from the present model assuming A = 10 and 20 MeV * fm
values in the range favoured in refs. 899).It is seen that to a great extent the observed spectrum could indeed be understood. However, it must be emphasized that the experimental data are at present in a state of very rapid change and also the theoretical binding energies are very model dependent as noted later. Therefore a more serious
comparison
is not justified
at this time, and table
1 would
only serve
as an example that the existing data and theory can be made to agree. In the table an identification is also made of some peaks with the energies of the ww and nq’ thresholds. This is based on the observation of the coincidence of energies and also the relative magnitude of the corresponding SU(4) Clebsch-Gordan coefficients “). Of course, to be a model this identification would need the rates for pp+ ooy and nn’-y to be calculated. The most outstanding feature of the above results is the narrow widths (or the small imaginary parts of the energy). There are two reasons for this smallness. Firstly, the imaginary part of the annihilation potential is energy dependent as pointed out above. Secondly, and perhaps more important, only part of the strong experimental annihilation in the scattering region is accounted for by the contribu-
599
J.A. Niskanen, A.M. Green / Nfi bound states
tions
included
admixtures
above,
i.e. only
up to 70%
which are not in the present
with
calculation.
the relevant
Nd*
Am and Ad
At first sight, the inclusion
of
these components might be expected to increase the above widths by a factor of 2-3, as in the scattering region. However, as stated before, these states should also be considerably suppressed for energetic reasons. It should be noted that without isobars the present model does not give any annihilation at all in the 13D,-13G3 state. To get direct annihilation in D-waves would require annihilation into two p-wave mesons (one could be the vacuum 13P,,) and one s-wave meson - a feature not yet included. In addition, also some possibly important annihilation mechanisms like the one in fig. lb, are so far completely neglected. Annihilation into two mesons could have important effects through its rather different energy dependence as compared with three meson final states 14). However, these should not be as drastic as the use of some phenomenological energy independent annihilation potential fitted above the NN threshold. After all, it should be remembered that in annihilation at rest the final states with two mesons contribute only lO-20% to the total absorption 15). To study further the model dependence of the results a more realistic energy dependence was incorporated into the meson widths. In the above, the widths were taken to be the constant values T(m’) obtained at the actual meson mass. Clearly this is an overestimate below the three meson threshold. Therefore, as in ref. ‘) an energy dependence was included in the form
s4: 3
T,(s)
=2
4T(mZ)
J
exp [
--$
($-s;)
1*
(3.1)
Here the gaussian wave function is used to give the form-factor effect and q3 is the phase-space factor typical for p-wave decays. The momentum q. is appropriate for a decay with is assumed to kinetic energy for the width
the (meson) c.m. energy 4s = tn. The energy available for the decay be equally shared among the unstable mesons once the three-meson and the stable meson masses are subtracted away. This prescription was referred to as T(B) in ref. ‘). Now the widths of the states go
down further by a factor of 3 or more (the circles in fig. 2), while the real parts of the energies remain virtually unchanged. In this way, for any state the widths will be small enough to be compatible with the experimental results. Even with the omissions of the isobar effects and of the two-meson final states mentioned above it is hard to see that these extremely small widths -now mostly in the order of l-10 MeV - would qualitatively change in a more complete calculation. Another parameter, in addition to A, which has some uncertainty - without being actually completely free-is the potential cut-off parameter r, of eq. (2.3). Here it is taken to be the same as the quark contribution to the proton charge rms radius and set at 0.6 fm - a number often used in the literature. Varying this by only 0.05 fm down to 0.55 fm - a reasonable lower limit - had a large effect on the binding energies making them much larger. The changes were largest in the tensor coupled
600
J.A. Niskanen, A.M. Green / NN bound states
states - amounting
to several
hundreds
of MeV - while in the uncoupled
states the
effect was “only” 50-100 MeV. However, the widths were not effected. It is interesting to note that a similar strong sensitivity on r, was not found in the annihilation cross section above the threshold. Of course, this dependence on r, is not necessarily a drawback, used to pin In ref. ‘) model and approximate
since the experimental spectrum - when better determined - could be down r, or, more generally, the real part of the NN potential. comparisons were made between the present separable annihilation the phenomenological Dover-Richard annihilation potential using equivalent local potentials, 1 V&local)
=G
j V,,,(r,
r’)$(r’)rr2
dr’.
(3.2)
In the estimates of ref. ‘) very simple forms for the wave functions were assumed and a qualitative agreement was obtained for the real parts with A = 20 MeV * fm. Also these real parts were found to be roughly similar to the meson-exchange part. It might be of some interest to remake this comparison with realistic bound-state wave functions. Therefore, in the 31S0 state the meson-exchange potential was switched off and the strength of the annihilation varied. The result, shown by squares in fig. 2, is that binding is now achieved only for h B 10 MeV - fm, and the binding energy equal to the one due to only the meson exchanges is achieved for A = 25 MeV * fm. This means that the equivalent local potentials obtained below threshold with the approximate wave functions are, in fact, in a reasonable agreement with a more exact calculation. It is also remarkable to see that rather narrow bound states can be produced by the annihilation potential alone! A more quantitative comparison between the present results and the DoverRichard annihilation potential is given in fig. 4. There the real part of the effective local potential is shown for the states 31S,,, 13S, and 33P, together with the real part of the Dover-Richard potential (dotted curve). Qualitatively the results agree reasonably well for rb 1 fm as found already in the approximate estimate of ref. ‘). The longer range and a kink at 1 fm are typical of the P-waves and are even more pronounced in the deeper bound states. Also for the S waves the deeper binding of the ‘3S,-‘3D1 case gives a longer range. The imaginary parts at Ea = 100-200 MeV are smaller by a factor of about 4 and & for the constant and variable meson widths, respectively. The wave functions r+(r) all have their maxima around r = 0.7-0.8 fm. The sharpness and height of this maximum is determined by the binding energy.
4. Conclusion In conclusion, it has been shown coupled channels NN+ MIM2M3 can give rise to deeply bound NN It is natural that coupled channels
that the quark rearrangement model leading to and eventually to a separable optical potential states with very narrow widths even in S-waves. approaches should have an energy dependence
J.A. Niskanen, 1.
601
A.M. Green / NN bound states
-V,,,(local) ‘, MeV
LOO 4
300 I-
200 I-
1oc l-
0.5 Fig. 4. The real part
1.0
1.5
r fm
of the effective local potential for various bound states (with compared with the Dover-Richard annihilation potential.
A = 20 MeV . fm)
such as to give smaller widths than an energy-independent optical potential as in ref. I’). For example ref. 16) gets small widths for some bound P-wave states and one D-wave, but S-waves are not considered. There the model for annihilation is based
on nucleon exchange giving a range -0.1 fm. However, the strength of the potential must be taken incredibly large - effectively making also the potential range considerable. It is hard to see the physical significance of such a potential, when one considers that the meson and nucleon sizes are already in the range OS-l.0 fm. A similiar short-range phenomenological potential with a linear energy dependence is given
in ref. 17). However,
this potential
- successful
in the scattering
region -
cannot be used in the present problem, since its imaginary part becomes positive already for binding energies of a few tens of MeV. In coupled channels this unfortunate behaviour does not arise. As pointed out earlier a more appropriate energy dependence for phenomenological approaches would be the exponential form proposed in refs. 9*‘o). The widths of the bound states depend on the treatment of the intrinsic widths r,,, of the final-state mesons (eq. 3.1) - an essential ingredient of the imaginary part of the NN annihilation potential below meson thresholds. Using a realistic energy dependence for r,,, reduces the widths of the bound states from a few tens of MeV
602
J.A. Niskanen,
A.M. Green / NN bound states
down to the region l-10 MeV even for S-states. This result is practically independent of the strength A, the free parameter of the model. The real parts of the binding energies (or the masses of the NN bound systems) vary nearly linearly with a moderate slope as a function of A. The sensitivity to the treatment of the innermost region of the meson-exchange potential is, however, quite drastic. With the present choice the agreement with the available data of refs. 596)obtained for A = 10 MeV * fm is very tempting, but should not be taken too seriously at the present stage. It is to be hoped that further experimental studies of these possible bound states will shed light on the real part of the NN interaction in particular. So far only the ground states have been studied. Due to the depth of the states ‘3P2-‘3F2,‘3S1-‘3D1and 13P0 it is plausible that their excited states might also exist. It may be worth noting that in ref. ‘) there was found a sharp peak at low energies for small h’s in the ‘3P2-13F2 state, which could be due to a quasibound state just above the NN threshold. The authors wish to thank Dr. C. Hajduk for useful correspondence Moalem for his criticism of the manuscript.
and Dr. A.
References 1) W.W. Buck, C.B. Dover and J.M. Richard, Ann. of Phys. 121 (1979) 47 2) C.B. Dover and J.M. Richard, Ann. of Phys. 121 (1979) 70 3) J.P. Ader, B. Bonnier and S. Sood, Phys. Lett. S4B (1979) 488; 1OlB (1981) 427; Z. Phys. C5 (1980) 85 4) A.M. Green and M.E. Sainio, J. of Phys. G6 (1980) 1375; A.M. Green, J. of Phys. G8 (1982) 485 5) a) P. Pavlopoulos et al., Phys. Lett. 72B (1978) 415 b) B. Richter et aI., Phys. Lett. 126B (1983) 284 c) L. Adiels et al., Phys. Lett. 13SB (1984) 235 6) G.A. Smith, Invited talk at the 7th Int. Conf. on experimental meson spectroscopy, Brookhaven, NY, April 1983 7) A.M. Green, J.A. Niskanen and J.M. Richard, Phys. Lett. 12lB (1983) 101 8) A.M. Green and J.A. Niskanen, Nucl. Phys. A412 (1984) 448 9) A.M. Green and J.A. Niskanen, Nucl. Phys. A430 (1984) 605 10) A.M. Green and J.A. Niskanen, Helsinki preprint HU-TFT-83-54, to be published in International review of nuclear physics vol. I, ed. T.T.S. Kuo (World Scientific, Singapore, 1984) 11) C.B. Dover and J.M. Richard, Phys. Rev. C21 (1980) 1466 12) L. Tauscher et al., Proc. IVth European Symp. on NN interactions, Santiago de Compostela, Spain 1982, Ann. de Fis. 79 (1983) 24 13) H. Poth et al., Phys. Lett. 76B (1978) 523 14) A.M. Green, J.A. Niskanen and S. Wycech, Phys. Lett 139B (1984) 15 15) U. Gastaldi and R. Klapisch, Proc. Int. School of Physics “Enrico Fermi” LXXIX, ed. A. Molinari (North-Holland, Amsterdam, 1981) p. 462 16) O.D. Dalkarov, IS. Shapiro and R.T. Tyapaev, ITEP preprint 21 (1984) 17) J. Cotd et al., Phys. Rev. Lett. 48 (1982) 1319 18) J.A. Niskanen, V. Kuikka and A.M. Green, Helsinki preprint HU-TFT-84-33 19) B.R. Karlsson and B. Kerbikov, Nucl. Phys. B141 (1978) 241