Int ig iPi5tiIj

This structure is obvious once we recall that tprii must transform like a vector (by analogy with ordinary spin), so its contraction with <j> will be a scalar.

9.9 ONE-PION EXCHANGE

We are now ready to calculate the one-pion exchange (OPE) diagrams in perturbation theory. Since this is the first time we have dealt with spinor fields, we will work out all the details carefully.

The basic structure of the second order result for non-forward scattering is identical to the results we worked out for 4? theory. The S-matrix is x T {\xl){xi)^Tii){xl)(t)i(xl)\-.-4>{x2)'i5T:ii>{x2)<t>J{x2): } |piSl,p2S2),

/here the sx are the spin quantum numbers of the nucleons, the two particle states

Throughout this discussion we will suppress the fact that the nucleon field i/> and the Dirac spinors u and v are really direct products of two-component spinors in isospin space and four-component spinors in Dirac space; the two-component isospin structure is implied but not written explicitly.

The evaluation of (9.113) parallels the evaluation of (9.84) quite closely, so the steps will be quite familiar. As in the earlier calculation, only the b and 6t terms from the field expansions will contribute; they are precisely what is needed are

|PlSl,P2S2) = 6pllSl&p2,S2|0) {P2s2^P'ls'l\ = (°l h'2AhP\>°\

and the field expansions are

9.9 ONE-PION EXCHANGE

to "balance" the two 6t's in the initial state and the two b's in the final state, which is required if the matrix element is to be non-zero. One new feature is the anticommutation relations satisfied by the annihilation and creation operators; in place of Eq. (9.85), we have the identity

x [-<5fcipi<5rlSl<5fc2p2^r2S2 + ¿fcipi^nsa^fcjp^rjsi] • (9.114)

To prove this, note that there are four possible pairings of b's with tf's and that the signs are different because of the anticommutation relations. For example, two terms with different signs result when is moved to the left in the following expression:

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