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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3. 7. Nuc1ear forces us<strong>in</strong>g the method of unitary transformation<br />
evaluated via eqs. (3.210) and (A.l) one gets contributions of the follow<strong>in</strong>g types:<br />
1. [II Lt]<br />
t<br />
v = 3 -3N + LVt<br />
t<br />
2. [II Lt]r,H" 77 [II Ls]<br />
t s<br />
v = 4 -3N + K, +<br />
L Vt + L v s<br />
t s<br />
3. [II Lt] 77AL,\ 4krr.+irr. H,,77 [II Ls]<br />
t s<br />
and [II Lt]r,H",\4krr.+imAlm77[II Ls]<br />
t s<br />
V = 4 - 3N + K, + 2km + im + Zm + L Vt + L Vs (3.228)<br />
t s<br />
4. [II Lt]r,AL,\4km+irr. H",\4kn+<strong>in</strong> A1n77[II Ls]<br />
87<br />
(3.226)<br />
(3.227)<br />
t<br />
V = 4 -3N + K, + Zm + Zn + 2km + 2kn + im + <strong>in</strong> + L Vt + L vs · (3.229)<br />
t<br />
Bere, v denotes the correspond<strong>in</strong>g power of Q. In case of the unity operator <strong>in</strong> eq. (3.197) one<br />
has to drop the v's. <strong>The</strong> effective potential can be easily read off from the effective Bamiltonian<br />
Via<br />
(3.230)<br />
In the follow<strong>in</strong>g seetion we will give concrete examples and calculate the lead<strong>in</strong>g orders of the two<br />
nuc1eon potential.<br />
3.7 N uclear forces us<strong>in</strong>g the method of unitary transformation<br />
We will now apply the formalism described <strong>in</strong> the last seetion and derive an effective Hamiltonian<br />
act<strong>in</strong>g on the purely nuc1eonic subspace of the full Fock space at lead<strong>in</strong>g and next-to-lead<strong>in</strong>g<br />
orders. As a start<strong>in</strong>g po<strong>in</strong>t we use the effective chiral <strong>in</strong>variant Hamiltonian for nuc1eons and<br />
pions [127] . It is based on the effective Lagrangian given <strong>in</strong> ref. [78].38 It reads:<br />
9A t �<br />
(3.231)<br />
-N TCf· ·'\17rN<br />
2j'lr<br />
1<br />
- 2j;<br />
(7r ' 8/17r)(7r ' 8/17r)<br />
+ _1_NtT . (7r x ir)N<br />
4j;<br />
'<br />
+ �CT (NtCfN) . (NtCfN) + �Cs (NtN) (NtN) ,<br />
j \ Nt (2c 1 M;7r2 - C3( 8 /1 7r . 8/17r) + C4 Cijk Eabc aiTa ('\1 j'lrb) ('\1 k'TrC)) N ,<br />
'Ir<br />
D1 (NtN) (NtTCf' :�7rN) + D2 [(NtTCfN) x<br />
4h<br />
8h<br />
2<br />
x (NtTCfN)]<br />
. '�7r<br />
38 For the contact <strong>in</strong>teractions with four nucleon legs we have used the set of terms given <strong>in</strong> appendix F.<br />
(3.232)<br />
(3.233)<br />
(3.234)<br />
(3.235)<br />
(3.236)