The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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