In this expression the gradient matrix element is given by, see Eq. (2.34)

We evaluate now the matrix elements of the two processes of figure 5.3

and processes (a) and (b) correspond respectively to the contractions (the + (-) sign is associated to the space (time) part),

according to the following recoupling diagrams, of figure 5.4 and figure 5.5, where we are using for the contraction symbols of figure 2.12 a double notation as in figure 2.4 or 2.5b to include isospin.

We finally get for the pion-nucleon matrix element of process (a) for example the expression (5.16). It is of course hermitic and one may check simply from the average value (5.15) that the matrix element (b) is the complex conjugate of (a).

In these expansions the integration over r brings in the momentum conservation requirements. Thus we have, see ref. [14]

Likewise the integrals with a gradient are explicitly given by,

(5.19)

(5.20)

(5.21)

3/2

3/2

g 1

(r) v

(r) v

(r)

=

2