The p meson

must interact with two pions in order to conserve 9 parity.

Here

2 Φ

(F) is the p field with substitution of its isospin wave function [1] by the operator [1]. The gradient must be a symmetrized operator in the two pion

μ

T

coordinates. The various possible processes are shown in figure 5.13.

Y X Y X

Figure 5.13

They give the same geometry since they yield invariant triple products in the operators. For example using the result,

we get for the various multipoles of the p field expressions the structure of which is of the form (x = 6, M, L)

(5.58)

with recoupling diagrams of the type given in figure 5.14.

P

Here we have made a change of notation to distinguish the and л creation operators we use C for the p mesons. Furthermore product of invariant ma

~

Thus we define the symmetrized function (with 812-1/2 if 22 or

12-1 if 11-22),

(5.60)

(5.61)

with a sum over the two lines if 11 22 for antisymmetrization purposes.

E1

) is

N

=

2

P

2E

x f (P2) 4p

1

LV

The w can interact with at least three pions for parity conservation and we adopt the following form for the interaction, where we have performed the scalar product .7 in isospin space (hence the replacement of the isospin vectors [1] by [1] in one of the boson field as explained in Eq. (5.3))

where the derivative must be symmetrized between the three pion coordinates. The evaluation of the energy goes along the same lines as above. We first note that the isospin part yields an invariant triple product of the operators in isospin space according to figure 5.15.

while the geometry of the angular momentum part is of the form shown on figure 5. 16.

where s is the proper symmetrization weight. Finally we can define the in-
ijk
teraction energy operators in a way similar to the NN interaction :