The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.8. Two-nuc1eon potential 101
Clearly, J2 (J4) is quadratically (quartically) divergent while Jo diverges logarithmically for large
momenta and p, is some quantity with the dimension of mass (renormalization scale). Note that
we could introduce other forms of divergent integrals, which depend on a single dimensionsfull
scale (denoted here by p,). Another choice of defining these divergent loop integrals would give
different finite subtractions but leads to the same non-polynomial terms in the potential. We will
comment more on that later on.
Consider first the one-Ioop corrections to the OPEP, eq. (3.282). The relevant divergent integral
lS /
d3Z ZiZj
(2n)3 w( = a Oij , (3.285)
where i, j = 1,2,3 and a is some (infinite) constant. No other two-dimensional symmetrical (in i
and j) tensors apart from Oij can enter the right-hand side of this equation, since the integrand
depends only on Z and the integration is performed over the whole space. Multiplying both sides
of eq. (3.285) by Oij and performing the summation over i,j we obtain
1/ d3Z Z2
a = - ---
3 (2n)3 w(
. (3.286)
Using eqs. (3.285), (3.286) and (D.1) we can now rewrite the OPEP (3.282) as:
4
. 2
V1�\-lOOP = 48�� 1: (Tl ' T2)(0'1 . if)(0'2 . if) �� {5M; + 3M; In :� + 4h -6M;Jo} , (3.287)
in terms ofthe two divergent loop functions JO,2 defined in eq. (3.284). Therefore, this contribution
has exactly the same form as the OPEP (renormalized OPE),
(3.288)
provided we redefine the coupling constant g� (the superscript "0" denotes the leading term in
the chiral expansion) in the following way:42
(gT)2 = (gO)2 _
{5M2 + 3M 2 In M; + 4J _
(g�)4
A A 12n2 fir 11: 11: 4p,2
2 11:
6M2 J }
0 (3.289)
Clearly, gA and g� differ by terms of second order in the chiral dimension. Consequently, all NLO
one-Ioop corrections to the OPEP can be taken care off by renormalization of g�.
In addition, there are the one-Ioop corrections to the lowest order four-fermion interactions shown
in eq. (3.283). Proceeding analogously to eqs. (3.285), (3.286) and making use of eq. (D.1) one
can express the divergent integral entering eq. (3.283) in terms of the loop functions JO,2 as
(2)
V NN,1-1oop -6::1 ; C� (3 - Tl . T2) (0\ . 0'2) / d3ZZ4 wi3 (3.290)
- g� j2 C� (3 - Tl ' T2) (0'1 . 0'2) {5M; + 3M; In M
4 � + 24n 4h -6M;Jo}
11: p,
or in a more compact notation
(3.291)
42 Note that this expression may change if one performs a complete renormalization (including the wave function
renormalization) .