The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

16 2. Low-momentum effective theories for two nucleons
x
+ +
• • •
Figure 2.2: The diagrammatic representation of the integral equation (2.6).
The shaded blob corresponds to the T-matrix and the filled circles denote the
effective potential.
where the regulator function JA(p 2 ) is 1 for p « A and 0 for p » A.lO We will now solve the
equation (2.6) analytically with the renormalized potential (2.7). We will not restrict ourselves to
a concrete choice of the function JA(p 2 ). The solution of the Lippmann-Schwinger equation can
be expressed as
T(p' ,p; k) �
IA(P"
) (�o P'''Yij(k)P'j) JA(p') . (2.8)
The number n depends on how many terms are kept in the potential (2.5). We will consider only
the first two terms in the expansion (2.5) and hence take n = 1. It is convenient to express the
potential v;.eg (pi, p) in the form
where the matrix )..ij is given by
The LS equation (2.6) can now be expressed in a matrix form like
with
I(k)
r(k) = ).. + )"I(k) r(k) ,
roo q
2 dq q2 fJ. (q2)
JO k
)
2 -q2 +ic
2 d q4 fJ.(q2)
00
Io q q kLq2+ic
(2.9)
(2.10)
(2.11)
(2.12)
lOThe function JA (p2) must decrease fast enough far large p to ensure that all ultraviolet divergences in the LS
equation are removed.