Figure 5.3. Some Feynman diagrams describing the most important meson-exchange processes which contribute to the NN interaction. Time goes upwards. Thin vertical lines: nucleons. Thick vertical segments: A resonance in an intermediate state.
short-range (h/mwc & 0.25 fm) repulsion and a sizable spin-orbit term. The contribution of one-8 exchange (h/msc & 0.2 fm) is relatively small, but is included in the model. The contribution of one-^ exchange is usually neglected because of its weak coupling to nucleons.
An important intermediate-range attractive component of the NN interaction is produced by two-pion exchange processes (h/2mnc & 0.7 fm). Some Feynman diagrams contributing to this component in the MEM are shown in Fig. 5.3 (diagrams (b)). Intermediate-state nucleons can be in an excited A-resonance state with m^c2 = 1232 MeV. This is the lowest excited nucleon state (spin 3/2, isospin 3/2). The first three diagrams represent the so called uncorrelated two-pion exchange processes in which pions do not interact strongly between themselves. In addition, one has to include diagrams where exchanged pions interact strongly with themselves, forming, e.g., hadronic resonances. An example is given by a right-most diagram (b) in Fig. 5.3, where exchanged pair of interacting (correlated) pions forms a vector (JP = 1+) p-meson.
Because the meson-nucleon coupling is strong, one should check the contribution of many higher-order processes for their importance in the NN interaction. It turns out that the most important are those involving simultaneous exchange of a pion and a p-meson. Two Feynman diagrams (diagrams (c)) describing these processes are shown in Fig. 5.3.
The summation of Feynman diagrams yields the scattering matrix which has to reproduce NN scattering data as well as the 2H properties. Using a fitting procedure, one determines with a high precision both the meson-nucleon coupling constants and the form-factors.
However, for the convenience of many-body calculation one is interested in representing the NN interaction in the form of a "potential" which would be equivalent to the NN field-theoretic MEM. This is usually done in the frame of the one-boson-exchange (OBE) model, where many-pion exchange contributions are modeled by the exchange of a scalar a-meson (m a C tt 550 MeV) with an appropriate aN coupling constant. This fictitious a meson of the OBE model reproduces a very important intermediate-range (h/mac tt 0.4 fm) component of the NN interaction.
In the OBE approximation one can represent the NN interaction potential as a sum of one-boson exchange contributions,
This potential can be written explicitly in a relativistically covariant way in momentum space,
The OBE potential is nonlocal in coordinate space. Acting on a NN wave function it gives
where r = ri — rj and r' = ri — rj. The interaction term depends not only on the relative distance between nucleons, but involves its neighborhood and a two-body wave function in this neighborhood.16
We may still represent it in an equivalent form of a formally local but momentum-dependent operator BE (r, q), where q = —ihVr. Using the well known property of the space-translation operator (see, e.g., §26 of Schiff 1968)
16To be contrasted with the standard case of a local interaction where (r |i)|°c | r') = v^oc(r)&(r'-r), so that (rVjc|^) = vjc(r)^(r) .
we can rewrite the term in the coordinate representation as
We see, that the non-local vi)BE is equivalent to a formally local momentum-
dependent Vi)BE(r, q). In the non-relativistic approximation, only the terms quadratic in momenta are retained. In this case, the spin and momentum structure takes a very familiar form, expressed via the operator invariants considered already in § 5.5.1 while constructing a phenomenological NN interaction. The pseudoscalar-meson component is a static (momentum-independent) operator involving the ai ■ aj, Sj operators alone. The scalar-meson contribution VOBE contains q2 and spin-orbit L ■ S operators. The richest spin and momentum structure results from the exchange of p and w vector mesons. This V)^ contains all previously listed operators. However, one should keep in mind that the OBE potential is par excellence a relativistic model. Therefore, its exact momentum dependence, that reflects the non-locality in coordinate space, is actually much more complicated.
An OBE model can very well reproduce existing NN data. Recent OBE models fit ~ 4300 pp and np scattering cross sections at collision energies below 350 MeV (in laboratory frame). The very high quality of fitting is similar to that reached for NN-interaction potentials constructed by the Nijmegen group (Stoks et al. 1994, %2/datum = 1.03) and the Argonne group (Wiringa et al. 1995, %2/datum = 1.09).
Many-body interactions arise naturally in the meson-exchange models: they are represented by Feynman diagrams which cannot be reduced to a sequence of NN interactions.
For instance, in diagram (a) of Fig. 5.2 the first meson exchange transforms a nucleon into a A resonance. This diagram does not describe an NN interaction which should have a nucleon pair in the final state. The second pion-exchange process starts with an NA pair,17 so that it is clearly not an NN —y NN process. The presence of the third nucleon is necessary for the whole process to occur, and therefore the diagram describes a genuine NNN interaction. Another example of a Feynman diagram contributing to the NNN interaction is shown in Fig. 5.2 (diagram (b)).
17 We remind that N = n or p, while N is an abbreviation for the word "nucleon".
Diagrams representing the three-body interaction contain an exchange of at least two mesons. The NNN interaction resulting from two-pion exchange becomes important in systems where three nucleons can be localized simultaneously within a range h/mn c & 1.4 fm. Therefore, NNN forces can be significant in nuclear matter at normal nuclear density. Their effect is also visible in the ground-state energy of much less dense few-nucleon systems like 3H and 4He.
Diagram (a) in Fig. 5.2 leads to a long-range component Vjl of the NNN potential of § 5.5.2. One can also consider the four-body interaction, generated, e.g., by meson-exchange processes described by Feynman diagrams (c) and (d) of Fig. 5.2. The longest-range component corresponds to the exchange of three pions (diagram (c)).
In practice, one theoretically calculates the long-range two-pion exchange component of the NNN force, but adjusts its strength to reproduce properties of A = 3 and A = 4 nuclei. In this way one gets the final form of Vj. The intermediate and short-range component of the NNN force is treated phe-nomenologically, as described in § 5.5.2.
Some examples of Feynman diagrams which represent meson-exchange processes and contribute to the four-nucleon interaction are shown in Fig. 5.2 (diagrams (c) and (d)). As we have discussed in § 5.5.2, there in no need to introduce the four-body force on the phenomenological level to describe the A = 4 nuclei. In contrast, the NNN force is clearly needed to correct for the underbinding of the A = 3 nuclei by the NN force alone and to make theoretical saturation parameters of nuclear matter consistent with experimental ones.
Experimental data on nucleon-hyperon (NH) and hyperon-hyperon (HH) interactions are scarce and rather imprecise, in sharp contrast with highly precise and complete NN data. Only a few points of NH scattering cross sections are available, while HH scattering data are absent. Many bound-state energies of hypernuclei containing a single A hyperon have been measured. These data are important for studying the AN interaction. Particularly useful are the data on the A = 3 and the A = 4 hypernuclei, because they allow one to check precise solutions of the three- and four-body problem. The data on |H, ^H, and A He hypernuclei can be used to test the NA interaction in vacuum. Models of in medium (effective) AA interaction can be tested in double-A hypernuclei, such as AAHe, i0\Be, and A3\B. As we will see, the AN interaction in hypernuclei is responsible for an important effect of the so called A-E conversion. Consequently, data on hypernuclei yield also an indirect information on the NE and AE interactions. Studies of jiHe give a valuable information on the NE interaction. All in all, one can experimentally test models of BB interaction for the baryon-pair strangeness S = 0, —1, —2.
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