D ec 1 99 9 Chiral 3 π - exchange NN - potentials : Results for diagrams proportional to g 4 A and g 6 A N . Kaiser Physik Department T 39 , Technische Universität München

We calculate in (two-loop) chiral perturbation theory the local NN-potentials generated by the three-pion exchange diagrams proportional to g 4 A and g 6 A. Surprisingly, we find that the total isoscalar central 3π-exchange potential vanishes identically. The individually largest 3π-exchange potentials are of isoscalar spin-spin, isovector central and isoscalar tensor type. For these potentials simple analytical expressions can be given. The strength of these dominant 3π-exchange potentials at r = 1.0 fm is 4.6 MeV, 2.9 MeV and 1.4 MeV, respectively. Furthermore, we observe that the spin-spin and tensor potentials due to the diagrams proportional to g 6 A do not exist in the infinite nucleon mass limit. In a recent work [1] we have started to calculate the static nucleon-nucleon potentials generated by the (two-loop) 3π-exchange diagrams with interaction vertices taken from the leading-order effective chiral πN-Lagrangian. In ref.[1] we restricted ourselves to the evaluation of four relatively simple classes of diagrams which in fact comprise all graphs with the common prefactor g 2 A /f 6 π. It was stressed in ref.[1] that the chiral 3πNN-contact vertex depends on the choice of the interpolating pion-field and therefore one has to consider representation invariant classes of diagrams (classes I and II in ref.[1]) by supplementing graphs involving the chiral 4π-vertex. Another obvious consequence of this is that there is no unique way (in chiral perturbation theory) of defining the so-called " pion-nucleon form factor " which is often introduced in phenomenological models. The only meaningful separation of the interaction potential is the one into point-like 1π-exchange and 3π-exchange. The four classes of chiral 3π-exchange diagrams evaluated in ref.[1] gave rise only to isovector spin-spin and tensor potentials. The strength of the resulting coordinate-space potentials was very weak with values of ±0.1 MeV and less at a internucleon distance of r = 1.0 fm. Compared to the chiral 2π-exchange potentials [2, 3] these are negligibly small corrections. The purpose of this work is complete the calculation of the chiral 3π-exchange NN-potentials and to present results for the diagrams carrying independent prefactors g 4 A /f 6 π and g 6 A /f 6 π. We will see that a grouping of these diagrams into five different classes is very useful. Let us begin with some basic definitions in order to fix our notation. In the static limit and considering only irreducible diagrams the on-shell NN T-matrix takes the following …