Paris potential

Fig. 2. Phase-shifts of NN scattering for the (a) Pi,(b) and(c) partial wave. Predictions are shown for the Bryan-Scott (B-S) potential of 1969 [12] (long dashes), Nijmegen potential [13] (short dashes), Paris potential [14] (dotted), and the Bonn full model [15] (solid line). The solid squares represent the energy-independent phase shift analysis by Arndt et al. [16].

In the final version of the Paris potential, also known as the parametrized Paris potential [14], each component (there is a total of 14 components, 7 for each isospin) is parametrized in terms of 12 local Yukawa functions of multiples of the pion mass. This introduces a very large number of parameters, namely 14 x 12 = 168. Not all 168 parameters are free. The various components of the potential are required to vanish at r = 0 (implying 22 constraints [14]). One parameter in each component is the nNN coupling constant, which may be taken from other sources (e.g., nN scattering). The 2n-exchange contribution is derived from dispersion theory. The range of this... 

Fig. 3. (a) Neutron-proton spin correlation parameter Cnn at 181 MeV. Predictions by the Nijmegen potential [13] (long dashes), the Paris potential [ 14] (dotted), and the Bonn full model [ 15] (solid line) are compared with the data (solid squares) from Indiana [18], The ( /datum for the fit of these data is 54.4 for Nijmegen, 3.22 for Paris, and 1.78 for Bonn. The experimental error bars include only systematics and statistics there is also a scale error of 8%. In the calculations of the all three error have been taken-into account [25]. (b) Same as (a), but at 220 MeV with the data from TRIUMF [19]. The z /datum for the fit of these data is 121.0 for the Nijmegen, 16.1 for the Paris, and 0.49 for the Bonn B potential [6]. In addition to the experimental error shown, there is a scale uncertainty of 5.5%. In the calculation of the all errors were taken into account [25]. 

Thus, by comparing the experimental NN scattering data with the predictions by the Paris potential, meson-theory is not put to a test. This comparison tests the phenomenological potential which is, indeed, consistent with the data. The satisfactory x of the fit of the data by the Paris potential (cf. Table 1) cannot be used as a proof that meson-theory is correct for the low-energy NN... 

Fig. 5. Two components of the Paris potential (a) the singlet central potential and (b) the (7 = 1, S = 1) tensor potential. The dotted line represents the result as derived from the underlying theory. The solid line is the parametrized Paris best-fit potential [14].

In microscopic nuclear structure calculations, the off-shell behavior of the NN potential is important (see Section 4 for a detailed discussion). The fit of NN potentials to two-nucleon data fixes them on-shell. The off-shell behavior cannot, by principle, be extracted from two-body data. Theory could determine the off-shell nature of the potential. However, not any theory can do that. Dispersion theory relates observables (equivalent to on-shell T-matrices) to observables e.g., nN to NN. Thus, dispersion theory cannot, by principle, provide any off-shell information. The Paris potential is based upon dispersion theory thus, the off-shell behavior of this potential is not determined by the underlying theory. On the other hand, every potential does have an off-shell behavior. When undetermined by theory, then the off-shell behavior is a silent by-product of the parametrization chosen to fit the on-shell T-matrix, with which the potential is identified, by definition. In summary, due to its basis in dispersion theory, the off-shell behavior of the Paris potential is not derived on theoretical grounds. This is a serious drawback when it comes to the question of how to interpret nuclear structure results obtained by applying the Paris potential. 

Fig. 7. Phase shifts of some peripheral partial waves as predicted by a field-theoretic model for the 2n exchange (solid line, BONN [15]) and by dispersion theory (dotted line labeled P 73 [21]). Both calculations also include OPE and one-to-exchange. The dotted line labeled P 80 is the fit by the parametrized Paris potential [14]. Octagons represent the phase shift analysis by Arndt et al. [23] and triangles the one by Bugg and coworkers [24].

Fig. 8. itp contributions versus phenomenology in (a) Dj.ib) and(c) Dj. The curves labeled 2it Paris and 27t Bonn represent the predictions by the Paris and Bonn model, respectively, when only the contributions from n, 2it, and cu are taken into account. Adding the phenomenological short-range potential yields the dotted Paris curve (parametrized Paris potential [14]). Adding the up contributions (Fig. 9) yieids the solid 2it + np Bonn curve (Bonn full model [15]). 

For very low energies, the effective range expansion applies and the quantitative nature of a potential can be tested by checking how well it reproduces the empirical effective range parameters. We show this in Table 2. It is seen that the Nijmegen and Paris potential reproduce the four 5-wave effective range parameters very badly in terms of the /datum, which is 95 and 169, respectively. In the case of the Paris potential, this is entirely due to a wrong prediction for a... 

MeV ( ,ab = 2q /M). The abscissa, k, is the variable over which the integration in Eq. (5) is performed. It is seen that, particularly for large off-shell momenta, the Bonn B potential is smaller than the Paris potential. However, notice also that at the on-shell point (q = k, solid dot in Fig. 12) both potentials are identical (both potentials predict the same Ei parameter). Thus, the Bonn B potential has a weaker off-shell tensor force than the Paris potential. Since the Bonn B and the Paris potential predict almost identical phase shifts, the Born term (central force) in the Si state will be more attractive for the Bonn B potential than for the Paris potential. 

Fig. 12. Magnitude of the half off-shell potential < S, F(q,fe)pD, > is held fixed at 153 MeV. The solid curve is the Bonn B potential [6] and the dashed curve the Paris potential [14]. The solid dot denotes the on-shell point (fe = q).

We have described the developments in the field of realistic NN interactions since the event of the Kuo-Brown matrix elements in 1966. It turns out that each of three models for the NN interaction currently in use, represents one decade of the past 30 years. The Nijmegen potential [13] is an excellent example for approaches typical for the 1960s, the Paris potential a representative of the 1970s, and the Bonn full model for the 1980s. Moreover, we could clearly reveal that with the development of more consistent and comprehensive meson-models over that period of time the quantitative explanation of the NN data has continuously improved. This fact is one of the simplest and best arguments for the appropriateness of meson models in low energy nuclear physics. 

Paris potential. Cottingham et aL (1973) and Lacomhe et al. (1975) have derived a nucleon-nucleon potential which includes one pion and correlated and rmcorrelated two-ru and to meson-exchange potentials. The theory gives a realistic description of the interaction in the... 

The overall description of NN scattering and deuteron data with the Bonn and parameterized Paris potentials is very good. See Table 2.4 for some deuteron data. 

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