Hyperfine Structure of the Metastable Triplet State Helium-3

Abstract

A NONRELATIVISTIC derivation of the two-particle potential between two fermions, due to interaction with a boson field, was outlined by Tamm 1 and Dancoff 2 and generalized by Levy, 3 including higher-order terms in the interaction potential. Following Levy's treatment, the interaction terms of higher order can be represented by nonrelativistic graphs, distinguishing in addition the sequence of transitions and intermediate states. In this treatment, it does not seem to be understandable why the strongly diverging vacuum graphs are not considered, although according to the derivation they should appear, because the influence of them on the energy denominators is such that they are not separable. Further it is impossible to recognize the other divergent parts of the graphs as terms of mass and charge renormalization. This is impossible because the relativistic invariance has been destroyed in the very beginning of the derivation of the onetime formalism. In these cases Levy goes back to the Bethe-Salpeter formalism, 4 showing an approximate correspondence of the two formalisms in two special examples and presuming it for the whole. FIG. 1. Example of a Levy graph with w=2 and m-0. The energy denominators areAEi =£<0(p+ki) +£< l >(P+ki-r-k 2 ) -r-a>(k 2 ),run into the wave function a(p') of (1). The different graphs should be distinguished by the number of meson lines (n), the number of closed loops (m), and the different topologies r which are possible with given n and m. Further, we have to distinguish the order of the 2n points where the meson lines end. In this way, every Feynman graph (nmr) consists of (2n) I ordered graphs, built up by all permutations (71-) of the points without alteration of the topology. Our interaction operator consists of the sum over all these graphs,where for every term V. of the general formthere is an analytic representation(4)The meaning of the terms of this formula iswhere g is the coupling constant of the meson field, and fj. is the mesonic mass. Every point of the graph means a transition to another virtual state; E ex is the energy of the virtual state and depends on the lines lying between the two points. If two fermion lines of such a virtual state are both in the initial state or in the final state, the energy of both is W (see Fig. 1). All other fermion lines give E(p) = (mi,2 2 +p 2 )*, and the meson lines (a(k) = (/i 2 +k 2 )*. Graphs that contain a virtual state with only two fermion lines are to be cut out. The connection between the variables p" and k,, is given by the graph in the usual way. TheJT's have the same meaning as in Levy's paper. 3 The lines running backwards are related to negative energy states, the lines with normal direction are related to positive energy states. / in (3) is the number of lines running backwards. If a graph contains no closed loop (m = 0), the integrations go over all meson variables dk v . If it contains one closed loop (m-1), one variable k" of the adjacent meson lines can be eliminated, because 2k,,=0 holds for those lines. But here ...