The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

98 3. The derivation o{ nuc1ear {orces {rom chiral Lagrangians
Further corrections arise from irreducible one-loop diagrams 1-8 in fig. 3.9:
where
vY 211", ) 1-1oop, Irr. .
(3.272)
We have also calculated the contributions from various irreducible one-loop diagrams 1-8 in
fig. 3.10 and 1-2 in fig. 3.13, which involve self-energy insertions and vertex corrections. These
have not been considered in [74], [76]. The corresponding contributions are given by
where
V(2) 111",1-1oop,Irr. .
and by
V(2) .
NN,1-1oop,Irr.
4 3 { � �
g A / dl T1'T2 l .q( ( � l �) ( � �) ( � �) ( � l �))
(2111")4 (27r )3 wlw� � 0'1' 0'2 . q + 0'1' q 0'2 '
(� + �) [2 (0\ . q) (eh , q)} , (3.274)
Wz wq
p,ur.
(3.275)
(3.276)
We remark an unpleasant feature of V1(2)1_100 . . With t = _q2 one can rewrite the term
7r,
proportional to l/w� as (M; -t)-3/2. This function has a cut starting at t = M;. Physically,
this does not make sense since all cuts should be produced by multi-particle intermediate states.
We co me back to this later on.
Finally, the corrections arising by explicitly keeping the nucleon kinetic energy in one-pion exchange
tree diagrams with both vertices from the leading order Lagrangian represents the energydependent
part of the N N potential [76] given by
V(2) _
g
E - (21
2 E _ l (f 2 + l(2)
11" q2 + M;
A ( �
)2 Tl . T2 0'1' q 0'2 ' q
�) ( � �) m 4
( )3/2 '
(3.277)