Fortschritte der Physik / Progress of Physics: Band 26, Heft 4 [Reprint 2021 ed.] 9783112519080

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCIILUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 4 • 1978 • B A N D 26

A K A D E M I E

- V E R L A G EVP 1 0 , - M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung f ü r fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle K U N S T U N D WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, Postfach 160, oder an den Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der Physik** Herausgeber: Prof. Dr. Frank T^"""1'1"'1", Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Hitachi, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, DDR -108 Berlin, Leipziger S t r a B e 3 - 4 ; Femruf: 22 36221 und 2236229; Telex-Nr. 114420; Bank: Staatsbank der DDR, Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, DDR - 104 Berlin, Hessische Straßo 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutsohen Demokratischen Republik. Gesamtherstellung: VEB Druckhaus „Maxim Gorki", DDR - 74 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik** erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band: 180,— M zuzüglich Versandspesen (Preis für die DDR: 120,— M). Preis je Heft 15,— M (Preis Tür die D D R : 10,— M). Batellnummer dieses Heftes: 1027/26/4. © 1978 by Akademie-Verlag Berlin. Printed in the German Democratio Republic. AN (EDV) 57618

ISSN 0 0 1 5 - 8 2 0 8 Fortschritte der Physik 28, 2 1 5 - 2 4 0 (1978)

The Nucleon-Nucleon Interaction1) J . J . DE SWART a n d M . M . NAGELS

Institute far Theoretical Physics, University of Nijmegen, The Netherlands Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Introduction ! 215 The Quark Model as a Guide from "Field Theory" to Relativistic Quantum Mechanics. . . 216 Bethe-Salpeter Equation • . 220 BSLT-Equations or Relativistic 3 Dimensional Scattering Equations 222 Lippman-Schwinger Equation 223 Potentials in Momentum Space 224 Scalar Meson Exchange 226 Potentials in Coordinate Space 228 Reggeon Exchange Potentials 229 Field Theory, Dispersion Techniques, and the Quark Model 232 The OBE-Model 234 Comparison with Experiments 235 Different Fits to the Data 236 References 239

1. Introduction

The nucleon-nucleon interaction can be described in a variety of ways. First of all there are the purely phenomenological potentials e.g. the Reid potential [lie 68] or the separable potentials [PI 76]. We shall not have time to discuss any of these approaches. Second of all there are the "field theoretical" and the dispersion theoretical approaches [Br 76] to the problem. In these more theoretical potentials one can distinguish two different types of parameters. Firstly there are the coupling constants which in principle can be obtained from and/or checked in analyses of other types of experiments [Na 76]. Secondly one has the more phenomenological parameters mostly necessary to describe the inner region of the potentials. These latter parameters like core radius, height of soft core, etc. are up to now mainly fudge parameters to describe our ignorance. When one studies the field theoretical and dispersion theoretical approaches as used by different groups, then one must conclude that there is no essential difference between these approaches. The languages are different, one approach has at one place some advantage, the other approach at another place. At present the real difference between the different approaches lies in what type of exchanges are or are not included and how this is done. These differences are already so large that the differences between the methods in Invited talk given by the first author at the European Symposium on Few Particle Problems in Nuclear Physics, Vlieland, The Netherlands, September 1976 16

Zeitschrift „Fortschritte der Physik", Heft 4

216

J . J . DE SWART a n d M . M . NAGELS

obtaining the potentials cannot be judged. Differences in the off-shell character of the various approaches are at this moment probably impossible to assess. We shall describe mainly the "field theoretical" approach to the NN interaction with the stress on the OBE-potentials. Occassionally remarks about the other approaches will be made. Finally we shall look at several of the potentials available at present. From this comparison we exclude all approaches for which no decent comparison with the experimental data is made. I t is interesting and sad at the same time to read the literature. Many things have been "proved" and/or "disproved". Mostly in model calculations with unrealistic models. Especially about the meson-nucleon coupling constants a lot of nonsense is available. SU(3) together with some types of experiments analyzed in a sloppy and/or prejudiced way seems to produce the wildest results. Some of these are quoted as the truth over and over. The moral of these last remarks is: "Believe nobody" [iSw 3I\. 2. The Quark Model as a Guide from "Field Theory" to Relativistic Quantum Mechanics In the quark model [Ko 69] the nucleons consist of three quarks and the mesons of a quark and an anti-quark. The quarks are colored, where the color is considered as the source of the strong interactions between the quarks. These forces are then due to the exchange of gluons between the quarks and anti-quarks. The gluons form a color octet of neutral massless vector particles. This whole system is described by a local renormalizable gauge-field theory called quantum chromodynamics. The nucleon-nucleon force should in principle be derived in quantum chromodynamics as the force between two threebody systems. This has not been done yet and we are forced to look at the more oldfashioned methods. I t used to be customary to describe the nucleons and the mesons by local renormalizable field theories in which some of these particles were considered as elementary and thought to be point particles (e.g. the bare nucleón and the bare pion). The standard calculations were done with the help of these field theories [Ho 60, Pa 70, Br 76]. However, if one ealizes the compositeness of the hadrons, then one realizes at the same time that several of the concepts of the standard field theories, when applied to the hadrons, need rethinking and modification. One falls then back on a more phenomenological description, which one could call relativistic quantum mechanics. This relativistic quantum mechanics is based on translational and Lorentz invariance. We think here of the full Poincaré group, including space inversion and time reversal. Due to the composite structure of both the meson and the nucleón the meson-nucleonnucleon vertex (fig. 1) is not a local interaction anymore. Therefore we cannot rely on

ik F i g . l : The NN-meson vertex

local renormalizable field theory to find out what to do. We must try to describe it phenomenologically in such a way that we have invariance under Poincaré transformations, baryon number conservation, conservation of isospin, etc. One can write down now many different vertex functions. As an example consider the pion-nucleon vertex. The vertex function r we could choose e.g. as

r^ = gPiYbt or r p v = A i y ^ r

217

The Nùcleon-Nucleòn Interaction

where M+ is the TT+ mass. In the old prescription using renormalizable field theories the y 5 -vertex (PS-coupling) was preferred over the y ^ - v e r t e x (PV-coupling) because the PS-coupling is renormalizable. Now we do not have such a prejudice anymore and our choice needs rethinking. Another consequence of the compositeness is that the coupling constants g P and /p are not constants anymore, but functions of the scalar variables of the problem. Therefore g = g{p2, p'2, k2). Here p and p' are the nucleon momenta and k the meson momentum. As long as we are in a situation in which the major contributions come from situations in which p and p' are almost on mass shell, we could propably approximate g~g(-M2,-M2,k2)=g(k2). We shall assume that this is the case. The coupling "constant" is then only a function of the meson momentum k2. In nucleon-nucleon scattering only spacelike (k2 0) valuesare important. I t is then customary to write g(k2) = 0(0) F(k2) where gr(0) is the coupling constant and F(k2) is a form factor. As possible forms for this, form factor one takes F{k2) = [An2/(An2 + k2)]n

or

F{k2) =

e^y

with n = 1/2, 1, 3/2 or 2. In practice one can roughly compare them via the relation A ~ A„IYn, which follows from the low k2 expansion. We would like to stress here several points: (i) This form factor is mainly due to the compositeness of the nucleons and the mesons. Therefore this form factor cannot be derived using a local renormalizable field theory for the nucleons and the mesons. (ii) The region over which this F(k2) is described by some simple phenomenological form is probably very restricted. Low energy NN scattering probes only a limited region of spacelike k2 near k2 = 0. In NN scattering one really determines the coupling constant g(0)2) One is unable to distinguish between the different functional dependences used for the form factor. Therefore one should not try to extrapolate to the pole position and one should not call and quote g = g(—m2) as the coupling constant. The value of g(—m2) is very sensitive to the assumed shape of the form factor. I t is well-known that the PS-coupling for the pseudoscalar mesons gives a very much stronger nucleon-antinucleon-meson NNP-vertex than the nucleon-nucleon-meson NNPvertex. This is not the case for PV-coupling. We would like to stress that these are not the only possible couplings for pseudoscalar mesons, but only the two simplest possibilities. This very strong NNP-vertex was in the past the source of many difficulties with meson theory [Dr 52, We 52, Br 53], Using qualitative arguments based on the quark model we would like to point out that the relation between the NNP- and NNP-vertex is not simple as suggested by either the PS- or PY-coupling, but very complicated. The NNP-vertex is probably much weaker than either the PS-coupling or the PV-coupling suggests. In the simple additive quark model [ K o 69], which can qualitatively explain many features of baryons and mesons, one introduces a simple phenomenological quark-quark2

) This does not refer to dispersion methods.

16*

218

J . J . DE SWABT and M. M. NAGELS

meson interaction Lagrangian. This leads then to the lowest order time ordered diagrams of fig. 2. We notice that in this simple model for the NNP- and NNP-vertices one must in the NNP-vertex annihilate two of the three quark and antiquarks in each baryon and antibaryon. This probably will suppress the NNP-vertex with respect to the NNPvertex. Another way to look at this problem is to compare the nucleon-meson vertex with the meson-meson vertex in the quark model. Next several time-ordered diagrams will be

Fig. 2. NNP and NNP vertices in the simple additive quark model

drawn, the time increasing from left to right. For the initial meson we shall take the ideal o meson, which is thought to be a pure /U state. The physical meson is almost pure o, but a slight admixture of uu and dd states is very probably present. From experiment it follows that the 4> -s- K K decay rate (despite its small phase space) is much stronger than the 4> 37c decay. _ Diagrams like 3 a allow for the 0-meson the decay 4>o K K , but forbid the decay H

Z3

q \ Tt.p,...

Fig. 3 b Fig. 3. Diagrams involving creation of only one qq pair describing the ® „ decay vertex (a) and the NN-meson vertex (b) with connected quark lines

N 0 37t because the initial strange quarks are not annihilated. The analogous N + vertex is forbidden for the same reason. Diagram 3 b allows vertices like NNTC, NNp, NNe, etc. Diagrams like 4a do not allow for 0 the decays K K or 0 - > 37i. The by the diagram allowed decays like 0 Tjo are energetically forbidden. The analogous mesonnucleon vertex of fig. 4b allows N N0O.

qorT

qorSN^O

Fig. 4 a Fig. 4 b Fig. 4. Diagrams involving creation on only one qq pair describing the 0 O decay vertex (a) and the NN-meSon vertex (b) with disconnected quark lines

Diagrams like 5 a allow the decays —3TT and be compared with the NN meson vertex. 0

4>0

K K . This vertex should perhaps

The Nucleon-Nucleon Interaction

219

Very popular in high energy physics is the Okubo-Zweig-Iizuka rule [Ok 63, Zw 64, Ii 66], which states t h a t diagrams like fig. 3 are dominant. This would give the right selection rules for the 0-decay. I t would allow the cj> -> K K decay and strongly suppress the 4* 37t decay. I t is not totally clear t h a t diagrams like fig. 4 a are suppressed, but

TE.P,.

V-TZ)

Fig. 5 a Fig. 5 b Fig. 5. Diagrams involving the creation of more than one qq pair describing the 0 decay vertex (a) and the simplest form of the NN-meson vertex (b)

t h a t in . meson vertex (fig. 5b) is suppressed? This would be very welcome, because it would explain the pair suppression hypothesis for the 7tN system. I n standard field theory the pair suppression hypothesis requires some rather accidental cancellations [Br 76]. How to calculate in relativistic quantum mechanics the nucleon-nucleon scattering matrix? The scattering matrix elements S f i can be expressed in the transition matrix elements Mfi by = -(2„)* W(Pt - Pf)

&= «/-««,

n = qixqf

=

qxk

and the scalars : q\ fc», g • ft = 1

(qf

-

qf).

The potential matrix V can be expressed in terms of a set of operators P { and the invariant potentials V^{Jc2, q2, (q •fc)2),which are real scalar functions. i

Possible forms for the operators

(«•*)*) Pi-

are:

Pi = 1

Pt = ( • k) {a2 • k)

P2 = a1- o2

P 5 = (ffi • n) {o2 • n)

Pz = y

+ 02) • n

P 6 = (ff! • q) (a 2 • q)

Pi = (q • k) [(ff! • k) (G2 • q) + (a1 • q) (a% • fc)] P 8 = (q •fc)[(Ox •fc)(