The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

30 2. Low-momentum effective theories for two nucleons
powers of the interaction V(p, ij), then one can show that A(p, ij) becomes infinite at each order
in V(p, ij) if both p and q approach A. For instance, the leading approximation for A(p, ij) is
given by
A( � ;;'l V (p, ij)
p,q =
(2.97)
} E q - E p '
from which this singularity can easily be read off. Note that also the linear equation (2.94) is not
well-defined if both p ---7 A, q ---7 A. We proceed by regularizing the original potential V (k', k). We
multiply it with some smooth functions f(k') and f(k) which are zero in a narrow neighborhood
of the points k' = A and k = A and one elsewhere. The precise form of this regularization does,
in fact, not matter [123]. For the actual calculations presented here, we choose
f(k)
f(k)
f(k)
f(k)
1 ,
1 ( Cr(k -A+a)))
"2 1 + cos
1 ( Cr(k - A - a)) )
"2 1 + cos
0,
b
b
'
'
for
for
for
for
kSA-a and k'2A+a,
A-aSkSA-a+b, (2.98)
A+a-bSkSA+a,
A-a+bSkSA+a-b,
with a and b parameters of dimension [energy]. This modification of the potential V in a onedimensional
case is depicted in fig. 2.3. The operator A(p, ij) based on that modified potential is
weIl defined far p, q ---7 A.
Further, we would like to quantify the effect of this regularization. The unregularized and regularized
T-matrices satisfy the following LS equations:
T(ql, q2) V( - � ) + / d3 k V(ql, ql,q2
k)T(k, q2)
E E + .
q2 - k 2E
f(qt)V(ql, q2)f(q2)
+ f(qt) / d3k V(ql,k)f(k)Treg�k, q2) .
Eq2 - Ek + ZE
(2.99)
(2.100)
Let us choose, for simplicity, b = o. Then the function f can be interpreted as the projection
operator in the moment um space
f = 1 -g, (2.101)
where 1 is the unit operator and 9 is the projector onto the moment um states A - a < p < A + a.
The projectors 9 and f satisfy the usual algebra g2 = g, f2 = f, fg = O. Subtracting eq. (2.99)
from eq. (2.100) and projecting this difference on the f-subspace we obtain the following integral
equation for f(tlT)f == f(Treg - T)f
f(tlT)f = -fVg GoTf + fVfGo(tlT)f . (2.102)
To simplify the notation we have used here the operator form. As in the usual LS equation,
f(tlT)f is a half-shell quantity. Note that the first term on the right-hand side of this equation
is of the order 0 ( a) beca use of the insertion of the pro j ector g. In general, each insertion of 9
leads to an additional power of a. The formal solution of this equation can be written as
f(tlT)f = - (1 - fV fGO)-l fV 9 GoTf . (2.103)
Assuming the existence of the inverse operator (1-fV fGO)- l with a finite norm for positive energies
one can estimate the upper bound for the quantity f(tlT)f. The most important observation