The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.8. Two-nuc1eon potential
2 3 4 5
6 7 8 9 10
Figure 3.19: First corrections to the NN potential with ß-excitations: irreducible
vertex corrections to the four-nucleon contact interactions. For notations see
figs. 3.6, 3.16.
restrict ourselves to the leading and next-to-leading orders. Then we need the following operators:
).1 A01], ).2 A01], ).4A01], ). 0 A21] and ).1 A21]. Note that the operators ). 0 AO,l1] and ).1 A11] vanish, as
discussed above. Obviously, we obtain the same expressions (3.241), (3.242) and (3.243) for the
zeroth order operators, where the energy denominators are modified appropriately:
(3.317)
(3.318)
(3.319)
Here, E is a sum of the pion energies and N ß mass splittings. For instance, E = W1 + W2 + 2ß
for a state with two pions and two deltas. For the remaining operators ). aAlT] at order l = 2 we
have
109
(3.320)
(3.321)
The effective potential at LO and NLO can now be obtained by straightforward calculations. We