The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

158 5. Isospin violation in the two-nucleon system
leading to
.. ..
.. ...
Figure 5.1: Resummation of the lowest order (NtN)2 contact terms ", Co.
1 ( . mp )
6 = 2i In 1 + 2 21f A .
+
• • •
(5.25)
Expanding both sides in the small moment um scale Q '" p (for the amplitude A one uses the
expansion (5.20)) one obtains:
(5.26)
where
1 ( mp )
60 = ---; In 1 + i-A-1 ,
22
21f
6 = 60 + 61 + ... ,
(5.27)
Up to now we have not yet incorporated pions into the effective theory. The tree contribution to
the amplitude due to the one-pion exchange is given by
2 -+
4f2 �2 M2
-+ -+ -+
gA 0"1 . q 0"2 . q
n q + n
Tl . T2 ,
(5.28)
which sc ales as 0(1) and thus occurs at the same order as an insertion of C2p2. Consequently,
pions can be treated perturbatively and the leading contributions to the amplitude due to the OPE
are given by graphs 2, 3 and 4 in fig. 5.2. The last graph in this figure depicts the contribution from
the contact interaction D2M;, which can be, in principle, absorbed by the Co vertex.u To study
at which energies such a perturbative treatment of pions would be no more adequate, the authors
of ref. [91] used the renormalization group technique. We will not repeat their arguments here
and refer the interested reader to the original publication. We only note that the scale entering
the low-momentum expansion of the amplitude turns out to be
161f l;
ANN = - 2 - '" 300 MeV .
gAm
This leads to the expansion parameter Q/ ANN '" Mn/ ANN '" 0.46.
(5.29)
To end this section we would like to write down the expressions for A�I)_A�V) obtained in ref. [91]:
11 However, one needs to keep this term explicitly in order to be able to perform consistent renormalization of the
dressed OPE in the fourth graph in fig. 5.2.