The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

Appendix C
Small scale expansion within the
projection formalism
In this appendix we would like to generalize the consideration of appendices A and B to include
the (intermediate) Ll-isobar excitations. As discussed in the text, the Ll is treated in a similar
way as the nucleons, i.e. as the light particle (but not massless as N) with the mass Ll = mf:l. - m1
and with both spin and isospin quantum numbers equal 3/2. Now we will generalize the results
obtained in appendices A and B to include also the states related to >.0.
We start with the dimensional analysis of the decoupling equation (3.206) projected by >.0 and
examine the chiral power v for all terms entering the left-hand side of this equation .
• >.0 HK,Tj
v = 4 - 3N + r;, 2: 6 - 3N . (C.1)
Note that here and in what follows, N corresponds to the number of baryons (also Ll's).
This inequality follows because the transition between purely baryonic states requires contact
vertices with four baryon legs or relativistic correction terms. The minimal possible value
of r;, for such interactions is 2.
The following possibilities are to be considered:
v = 4 - 3N + 2q + j + r;, . (C.2)
1. q = 0, j = 0
In that case we have r;, 2: 2, l 2: 2. Consequently, one obtains:
v 2: 8 - 3N . (C.3)
2. 4q + j # 0
In that case 2q + j 2: 1. Prom the inequality (A.5) we obtain
v 2: 6 - 3N . (C.4)
v 2: 6 - 3N . (C.5)
I This is because a trivial momentum dependence of the f:l.-field due to the nucleon mass is factored out, see
eq. (3.160).
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