Can We Measure an Annihilation Range of the Nucleon-Antinucleon System?

Transcription

1 Can We Measure an Annihilation Range of the Nucleon-Antinucleon System? The range of the annihilation of the nucleon-antinucleon system is intimately related to our idea of the internal structure of the nucleon. In the old days of a point-like "Yukawa" nucleon this range was assumed to be given by baryon exchange with ll.ann = n/2mc = 0.1 fm only. On the other hand, in any picture containing quarks as constituents of the nucleon, annihilation will happen if the quark confinement zones overlap and quarkantiquark fusion and rearrangement occurs. However, there exists a great variety of theoretical conjectures on the size of the radius of the confinement region of the nucleon ranging from negligibly small in the skyrmion model, 0.5 fm for the little bag, 0.8 fm for the cloudy bag to 1.1 fm for the MIT bag: 1 A natural question arising is: Can we measure the range of annihilation and in this way deduce a radius at which the quark degree of freedom sets in? A measure frequently used for the size of the nucleon is the radius of the absorptive disk derived from the high-energy proton-proton (pp) and proton-antiproton (pp) scattering. Figure 1, for example, shows a comparison of the elastic differential cross sections of pp and pp at ISR energies. The differential cross sections are characterized by a diffractive peak which is excellently described by scattering on an absorptive disk having the shape of a Gaussian profile function. 2 It is, therefore, customary to parameterize the differential cross section by drr/dt = a exp(bt), where b is called the slope parameter. This Comments Nucl. Part. Phys. 1986, Vol. 16, No. 2, pp /86/ /$15.00/ Gordon and Breach, Science Publishers, Inc. Printed in Great Britain 85

2 N > QJ." 9.,. pp 1000 e o PP ' $!') !J 1! Is= 31 GeV lx100)...,. t '!J L:J {!51!5 ' 1 ls : S)Ge.D lxloj E ' b I 0.1 % " Is= 62GeV I 0.01 'f,, t' t, GeV 2 FIGURE 1 Differential cross sections of pp and pp at ISR energies. parameter is surprisingly constant over a large momentum range: forppb = (12 ± 2) (GeV/c)- 2 intherangel0 3 GeV/c>PLab > 30 GeV/c and for pp b = (13 ± 1) (GeV/c)- 2 over a range which extends to even lower energies, 10 5 GeV/c > PLab > 1 GeV/c. 86

3 From the slope parameter and assuming the Gaussian profile function one can determine the e-value of the interaction radius. It is the same for pp and pp scattering at high energies and has a value r = lin/25 = (1.0 ± 0.1) fm. At high energies the slope parameter is determined by the total inelastic cross section of which the annihilation is only a fraction. The constancy of the slope parameter b in the pp interaction down to PLab = 1 GeV/c, where the annihilation is the only inelastic channel, indicates that the range of the annihilation is about the same as the range of the interaction in high-energy hadron-hadron collisions. It is, however, not really clear how to deduce the range of the annihilation from this slope parameter. For this one would need a model to relate the profile function to the nucleon structure. In particular, it is not clear whether the absorptive disk represents a folded distribution of quarks and antiquarks or whether other constituents such as soft gluons or mesons do essentially contribute to the inelastic processes. A hint to the solution of this question comes from Fig. 2 which shows the differential elastic distribution for pp at the relatively low momentum of 1.6 GeV/c. 3 We see again a diffractive peak with b = 13 (GeV/c)- 2. However, at larger angles (resp. four momentum transfer t) a cross section much larger than the diffraction would give is measured. A partial wave analysis gives a maximal contributing angular momentum of /max = 13 which can be converted into an interaction radius via r = Iii/ PLab = 1.6 fm. At this momentum the distribution seems to be characterized by two radii, one of an absorbing disk producing the diffractive peak and a second larger one which we might intuitively assign as arising from boson exchange or, in terms of radii, to the size of the pion cloud. It is evident that at even lower energies where only a few partial waves contribute, the effect of the long-range interaction will become enhanced. In an experiment at the low-energy antiproton ring (LEAR) at CERN a Heidelberg group has measured the pp cross section down to antiproton momenta of 180 MeV/c or a kinetic energy of 20 MeV in the laboratory system. 4 In Fig. 3 the differential elastic cross section of pp at 287 Me V/c is compared to that of pp. At this momentum the pp scattering is 90% s-wave whereas we clearly see a contribution of higher 87

4 N 10 3,, > Cl) t:) E 10.., "O.... ""-" "O GeV/c pp '...,,,_.,,,,,.,,,,...,,,.,, _ '---' ' o t (GeV/c/ FIGURE 2 Differential cross sections of pp at PLao = 1.6 GeV/c. partial waves in the pp case. A quantitative shift analysis yields 50% s-wave, 40% p-wave and 10% d-wave composition. Let us first demonstrate that this difference is the consequence of the annihilation of the pp system. At low energies it is usual to discuss scattering in terms of the optical model potential. The exact solution of the scattering amplitude in this case is 5 r f 1 (0) = -2µ/li 2 (2l + l)pi(cos0) V(r)Rt(r)Mkr)r 2 dr (1) where R 1 (r) is the radial part of the exact solution of the scattering 88

5 ,,..._.._ l/l... c L. 0.CJ..., 287 MeV/c 10-1 pp 0 pp E 10-2 u c 'O... b 'O cm (degrees) FIGURE 3 Comparison of the differential elastic scattering of pp and pp at 289 MeV/c. The solid curve shows a best fit with an optical potential as described in the text. problem for the angular momentum l, j 1 (kr) is the spherical Bessel function and V(r) is the optical model potential with, in general, a real and an imaginary part. The real part of the pp potential is usually derived from the pp one-boson exchange potential by applying the G parity transformation. 6 For the long-range potential which is dominated by twopion exchange averaged over spin and isospin one does not expect much difference in the two cases. On the other hand, in the oneboson exchange model, w exchange is responsible for the strong short-range repulsion in the pp case. In the pp case where w has G = - 1 one would expect a strong attractive potential at short distances. In first-order Born approximation replacing R 1 (r) by 89

6 Mkr) ex (kr)' in Eq. (1) one realizes that the change of the real potential at small distances will influence the p-wave phase shift less than the s-wave. At 180 MeV/c the wavelength of pp in the center of the mass system is A. = 8 fm and k = 0.8 fm - 1 Therefore, even drastic changes of the real potential will not greatly increase the p-wave contribution at the low momenta considered here. The situation changes, however, if one introduces the strong imaginary part of the optical model potential required to simulate annihilation. The interplay between a long-range real part and a shortrange imaginary part determines that s- and p-wave contributions are about equally strong. It is natural to attempt to fit the pp differential cross section data with a simple optical model potential by varying the depth and the range parameters of the real and imaginary potentials only. With the range and depth of the real part the same as for the pp system at low energies, i.e., V 0 = 46 MeV, R = 1.9 fm, a = 0.2 fm for Wood-Saxon shape and the imaginary part of the Gaussian profile function of the absorptive disk discussed above, the solid curve shown in Fig. 3 is obtained. Unfortunately, potentials do not provide a direct physical insight. This is one of the reasons why the concept of scattering length and "effective ranges" were introduced 7 in the early days of nucleon-nucleon scattering. In the context of our discussion of ranges we wish to relate the potentials to the ranges of particular interactions. In the annihilation channel this seems natural (see, for example, Ref. 5) because the total flux of particles S removed from the incident wave is derived from the equation of continuity and reads: S = L Sn da = -2/h L"' Im V(r)P(r)d 3 r (2) where P(r) = t!j*(r)t!j(r) is the position probability density and Sn(r) is the probability current density. It is therefore natural to take the quantity ptb (r) = 2/h Im V(r)R((r)R 1 (r)r2 (3) as the absorption probability for the partial wave of the angular 90

7 momentum I. Since in our case all absorption goes into annihilation the quantity can be interpreted as a measure of the annihilation probability. The radial distribution of p7nn (r) will therefore indicate how the flux disappears by annihilation as a function of the relative distance between antiprotons and protons. Figure 4 shows the annihilation probability p7nn for the best fit parameters for the 287 MeV/c elastic angular distribution for I= 0 and 1. The absorption is well localized and has its maximum at 1.2 fm. As expected, the pp scattering fits do not depend strongly on the particular choice of the potentials (Gaussian, Wood-Saxon with different surface diffuseness) as we are dealing only with partial waves up to I = 2. All good fits reproduce closely the annihilation probability as shown in Fig. 4. For the "range of the nuclear force," i.e., the range of the real part of the potential V(r), the definition of a "range" is less evident. However, since the first nucleon-nucleon scattering experiments one introduced the notion of a "range of the nuclear force" as the >IE a>......,,, -'- 40 II).Q <Uw a. 20 I 0 "' r (fm) FIGURE 4 The absorption probability prb (r) = ImV(r)Rt(r)j 1 (kr)r2 as a function of the radius (distance between proton and antiproton) for s- and p-wave contributions. 91

8 radius of the square well potential giving the correct scattering amplitude at low energies. 7 As already pointed out the pp scattering is reproduced by an optical potential with a real part which has very similar parameters as the nucleon-nucleon potential. Consequently the range of the real part of the pp interaction is 2 fm and is the same as the range of the nucleon-nucleon force. The notion of two different ranges in the pp interaction is not new. Several theoretical papers contain this observation implicitly or.explicitly 8 based on an extrapolation to low-energy data. The recent Heidelberg experiment 4 covers the relevant energy range down to very low momenta. The interpretation of the low-energy data fits nicely into the interpretation of pp scattering at high energies. The low-energy scattering is described as wave scattering from a potential (kr = 1) whereas for the high-energy scattering a description as diffractive scattering from an absorptive disk is appropriate. However, one may intuitively identify the strong absorbing potential at low energies with the strong absorbing disk at high energies. There is a remarkable correspondence between this radius of the absorbing disk and the range of the annihilation potential derived from the low-energy scattering. Though the disk radius measures something different than the "range of the annihilation" discussed above the underlying physical process must be related. At very high energies the scattering is determined by quark-quark (and gluon) interactions with a range given by the profile function. The real part of the interaction can be neglected. At low energies the quark interaction range manifests itself by annihilation, i.e., an overlap of the confinement regions. However, for low energies the longer ranged real part which represents exchanges of color singlets, i.e., bosons, is clearly felt. We believe that it is possible to distinguish two interaction ranges in the low-energy pp interaction experimentally: one for the annihilation of about 1.2 fm and one for the nuclear force of about 2 fm. This interpretation fits in a natural way the high-energy results. Annihilation seems to be the only phenomenon at low energies which marks the appearance of the quark-quark interaction explicitly. It is up to theoretical models to eventually connect these phenomenological parameters to the internal structure of the nucleon. 92

9 BOGDAN POVH Max-Planck-Institut fur Kernphysik, D-6900 Heidelberg, Federal Republic of Germany THOMAS WALCHER Institut fur Kernphysik, Universitiit Mainz, D-6500 Mainz, Federal Republic of Germany References 1. "Skyrmion": T. H. R. Skyrme, Proc. Roy. Soc. A 260, 12 (1961); E. Witten, Nucl. Phys. B 222, 433 (1983). Cloudy bag: G. A. Miller, A. W. Thomas and S. Theberge, Phys. Lett. 918, 192 (1980); S. Theberge, A. W. Thomas and G. A. Miller, Phys. Rev. D 22, 2838 (1980); ibid. 24, 216 (1981). Little bag: G. E. Brown and M. Rho, Phys. Lett. 82B, 177 (1979). MIT bag: A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, Phys. Rev. D 9, 3471 (1974) For a recent review see R. Castaldi and G. Sanguinetti, CERN-EP/85-36, to be published in Annual Review of Nuclear Science. 3. F. Eisenhandler et al., Nucl. Phys. B 113, 1 (1976). 4. W. Bruckner et al., Phys. Lett. ISSB, 180 (1985); W. Bruckner et al., CERN-EP/ (submitted to Phys. Lett. B). 5. L. I. Schiff, Quantum Mechanics, Third Edition (McGraw-Hill, New York, 1968). 6. R. A. Bryan and R. J. N. Phillips, Nucl. Phys. B 5, 201 (1968). 7. J.M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (Wiley, New York, 1952). 8. C. Dover and J.M. Richard, Phys. Rev. C21, 1466 (1980). For a recent summary see, e.g., A. M. Green and J. A. Niskanen, in "Quarks and Nuclei," Int. Review of Nuclear Physics, Vol. 1, ed. W. Weise (World Scientific, Singapore and Philadelphia, 1984). 93