The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

100 3. The derivation o{ nuc1ear {orces {rom chiral Lagrangians
respectively. Note that Vl�: 1-1oop,red. again contains a term rv l/wg. Combining eqs. (3.274) and
(3.280), this unphysical contribution vanishes. Summing up the corrections eqs. (3.274), (3.276)
corresponding to irreducible graphs, which are again the same as in old-fashioned time-dependent
perturbation theory and those (3.280), (3.281), corresponding to "reducible" diagrams, one gets
the complete result, representing the one-Ioop contributions, which involve self-energy insert ions
and vertex corrections within the projection formalism:
Vb,1-1oop
(2)
VNN,1-1oop
(2)
The complete potential at NLO in the projection formalism is given by the express ions (3.270),
(3.279), (3.282) and (3.283). The formal expressions for the NN potential shown above contain
ultraviolet divergences. In the next section we will show how such ultraviolet divergences can be
removed by redefinition of the coupling constants.
Let us briefly discuss the NNLO potential given in eq. (3.255). The related diagrams are shown
in fig. 3.15. In that case no purely nucleonic intermediate states are possible. Correspondingly,
one obtains the same contributions in time-ordered perturbation theory and in the projection
formalism. This is also evident from the explicit form of the operators in eq. (3.255). The formal
express ions of the NNLO potential were presented in [78]. In the next section we will show the
renormalized contributions given by Kaiser et al. [108].
Finally, let us point out once more that the crucial difference between the two formalisms, timeordered
perturbation theory and method of unitary transformation, results in the treatment of the
energy-dependent term eq. (3.277). As already noted ab ove , it does not appear in the method of
unitary transformation. We note that many of the results derived here have already been found in
[108], where one- and two-pion exchange graphs were calculated by me ans of Feynman diagrams
and using dimensional regularization. The potential was applied to N N scattering in peripheral
partial waves (with orbital angular momentum L 2: 2), where it does not need to be iterated.
Our aim is to apply this potential to all partial waves and to the bound state problem. We will
comment more on that in the next chapter.
3.8.2 Renormalization of the N N potential at NLO
The effective N N potential at NLO derived in the last section is not well-defined, since it contains
ultraviolet divergent integrals. Here, we would like to show how to renormalize this potential.
Since we are dealing with an effective field theory, we can use the standard order-by-order
renormalization machinery for the Lagrangian, which was first developed in the context of chiral
perturbation theory. All these divergent contributions (at NLO) can be renormalized in terms of
three divergent (momentum space) loop functions,41
/00 dl
Jo = (3.284)
J1 T'
41 Clearly, all ultraviolet divergent integrals here and in what follows have to be understood as the limits A -+ 00
of the corresponding integrals over the finite range of momenta 0 < I < A.