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