The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

112 3. The derivation of nuclear forces from chiral Lagrangians
moment um dependence is due to the loop integration ensuring that the result is given solely by a
power-Iaw divergent integral. Note that no analogous diagrams with purely nucleonic intermediate
states contribute to the effective potential in our approach.
We now discuss the most interesting type of correction, the two-pion exchange diagrams with
.6.-excitations. Different to the purely nucleonic case, one has no "reducible" graphs and, consequently,
obtains the same results with time-ordered perturbation theory and the projection
formalism. The TPE graphs with the .6.'s at NLO corresponding to eq. (3.323) are shown in
fig. 3.22. The tri angle diagram 1 in this figure is related to the second, third and fourth terms in
the first line of eq. (3.323). The first term in the second line and the last term in eq. (3.323) refer
to graphs 2-5 in fig. 3.22.
We will not calculate the complete NLO potential since the inclusion of the .6. is performed,
basically, to interpret the physics associated with the low-energy pion-nucleon constants. Such a
calculation has been performed by the Munich group [109]. They have found that all one-Ioop selfenergy
and vertex corrections with .6.-isobar excitations lead only to mass and coupling constant
renormalization and have the same structure as the OPE and N N contact interactions. The only
new interaction in the Hamiltonian, that includes the .6.-field and is relevant for calculation of
the NLO potential is
where the 2x4 spin and isospin transition matrices Si and Ti are normalized via
�
(28" - if."kO'k)
3
1 .
-(28·· - Zf."kTk)
3
1) 1) ,
1) 1) •
(3.324)
(3.325)
(3.326)
An explicit form for the matrices S and T can be found, for example, in ref. [189]. In eq. (3.324) we
adopt the same value for the coupling constant corresponding to the 7r N .6. vertex with one derivative
as in [109]. The TPE diagrams with .6.-excitation were also evaluated (but not renormalized)
by Ord6iiez et al. [78] using time-ordered perturbation theory. For completeness, we collect here
the pertinent formulae for the renormalized TPEP with .6.'s worked out by the Munich group
[109]. There are three distinct contributions .
.6.-excitation in the tri angle graphs:
with
s
L(q)
D(q)
.6.
E
V4M; + q2 ,
� In s + q
q 2M1': '
1 100 dJ.l
arctan
mt::. - m = 293 MeV ,
2M; + q2 _ 2.6.2 .
..6. 2M" J.l2 + q2
V J.l2 - 4M2
2.6.
1':
(3.327)
(3.328)
(3.329)
(3.330)
(3.331)
(3.332)