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
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
18 2. Low-momentum effective theories for two nucleons<br />
Let us now consider the on-shell T-matrix (2.16). <strong>The</strong> imag<strong>in</strong>ary part of its <strong>in</strong>verse is given by<br />
Here we have used the equality<br />
k2 _<br />
P i7r<br />
q2 - 2k<br />
i<br />
= -7rkm .<br />
2<br />
8(k - q) ,<br />
(2.22)<br />
(2.23)<br />
where P stays for the pr<strong>in</strong>cipal value <strong>in</strong>tegral. Thus, the imag<strong>in</strong>ary parts of the T-matrix <strong>in</strong><br />
eqs. (2.21) and (2.16) agree automatically. This has to be expected from the unitarity of the Smatrix.<br />
Before fix<strong>in</strong>g the constants Co and C 2 we have to def<strong>in</strong>e the low-momentum expansions<br />
of iX(k2) and of the pr<strong>in</strong>cipal value <strong>in</strong>tegral P(I):<br />
1 + hk2 + i 4 k 4 + O(k6) ,<br />
m (to + t 2 k2 + t4 k4 + O(k6))<br />
(2.24)<br />
(2.25)<br />
where fi and ti are some constants. No negative powers of k2 <strong>in</strong> eq. (2.25) are allowed s<strong>in</strong>ce<br />
otherwise P (fo oo dq fÄ ( q2) / (k2 - q2)) would not exist for k = O. Expand<strong>in</strong>g the <strong>in</strong>verse of the<br />
T-matrix (2.16) and compar<strong>in</strong>g the first two terms of this expansion with those <strong>in</strong> eq. (2.21) we<br />
obta<strong>in</strong> the follow<strong>in</strong>g two conditions for Co and C 2 :<br />
7rm<br />
2a<br />
7rmr<br />
4<br />
Solv<strong>in</strong>g the second equation leads to two solutions for C 2 :<br />
where<br />
C 2 = � ±<br />
2ah + 7rm<br />
13 hVi5<br />
D = (7r2m2 + 2am7r(2h - hh) + a2( 41l + rm7r h - 4mhto))<br />
For the constant Co one obta<strong>in</strong>s from eq. (2.26)<br />
C _<br />
Is (2ah + m7r)( -2a1j + 2ahIs + Ism7r ± 2Vi5Is)<br />
_ _<br />
0 - 12 3 D12 3<br />
(2.26)<br />
(2.27)<br />
(2.28)<br />
(2.29)<br />
(2.30)<br />
Let us now consider the conditions, under which real solutions for the constants Co and C 2 exist.<br />
For that we should require<br />
D > 0. (2.31)<br />
From the def<strong>in</strong>ition (2.13) we see that<br />
(2.32)<br />
where 0: is non-negative dimensionless constant. This simply follows from dimensional reasons.<br />
For the expansion coefficients tn, <strong>in</strong> <strong>in</strong> eqs. (2.24), (2.25) we have:<br />
1<br />
<strong>in</strong>