where the functions K and L are given in eqs. (2.41) through (2.46). The summain (3.7) extend of course over all the multipolarities of the tions K From the remarks of Chapter I section 2 we get immediately for the spin O "scatering" term, = rt ie[1],[1] [1] [ and similar terms for pair creation and annihilation contributions with the 1 11 +12 added phase (-) and (-) respectively. For spin 1/2 the term of (1.12) is immediate since is a good quantum number, while the spin term s requires a simple recoupling, + For spin 1 the expression is as simple as in the spin O case since is a good quantum number Note that in expressions (3.8), (3.9) and (3.10) we have set Ho= 1, cf. eqs. (1.11), (1.12) and (1.13) in conformity with our basic assumption of no anomalous moment. Furthermore, the truncated space of ref. [1], Chapter VI, does not allow for pair creation or annihilation terms in the static moments of spin 1/2 and spin 1 fields. The contribution to the elastic scattering form factor of the w meson through the vector dominance graphs of figure 4.1.a and 4.1.b is simply of the form, according to Chapter I section 3 where w() is the field, the expansion of which is given in eqs. I(3.222) μ (4.1) Fig. 4.1 The analogous expression for the p meson must include a projector on the neutral component p with the isospin amplitudes c[1] given by eq. (2.2) and where in (4.2) the dot implies a scalar product in isospin space. Thus the isospin wave function in each term of the p expansion is replaced as follows Apart from this isospin coupling of the creation (or annihilation) operators to the amplitude c[1], the w and p mesons yield identical expressions. Furthermore, the Fourier-Bessel transforms of the radial part of the field which appear in the matrix elements are, for example, of the general form (see expressions I(3.232) through I(3.236)) K [ r2dr jq(qr) w*J, (r) 6μx = [ r2dr | p2dp j ̧(qr) j ̧(pr) √ √ fvj(P) VJA V2E while the space geometry is obtained, introducing space polarization amplitudes a[1] The vector form factor for the w-meson dominance term is thus given by For the field, these expressions hold with a replacement according to 4.3, The evaluation of the quadrupole moment requires a special treatment, since the usual procedure, followed in Chapter III, involves an interaction of the electromagnetic field with a conserved vector current. In contrast, here the interaction energy arises from the emission or absorption of a neutral particle. Thus we shall start from the defining equation for the energy of a quadrupole moment (rank two tensor denoted by Q interacting with an electric field gradient where the tensor product implies contraction of the tensor indices. w (4.13) We have here introduced the unit tensor of rank 2, Q. Eq. (4.13) is valid as long as the gradient of the field is constant over the volume of the system. On the other hand the vector dominance interaction energy is given by (1.21). Hence we shall recognize the vector dominance quadrupole moment Q by equating these energies, |