The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

90 3. The derivation of nuc1ear forces from chiral Lagrangians
fact that there is no non-vanishing operator Al H2'f}. At next-to-leading order, 8-3N, one obtains
a more complicated expression for the potential:
Ve�-3N) = 'f}(H4+AbA4H2 + H2A4Ao +A�AlHl +HlAlA2
t l l 1 tl 1 + AOA H2A Ao - "2 AOA AO'f}H2 - "2 H2'f}AoA tl Ao
+ AbA\€Ao - 1AbAl AoE - 1EAbAlAo + A�AlwAo + AbAlwA2
+ AbAl HlA2 Ao + AbA2 HlAl Ao (3.251)
Itl tl Itl 1
1 - "2 AOA AO'f}AoA Hl - "2 AOA AO'f}HlA Ao - "2 AOA t l Hl'f}AoA t l Ao
1 - 1 t l 1 t l t l I
"2 HlA AO'f}AoA Ao - "2 AOA AO'f}AoA wAo - "2 AOA t l wAO'f}AoA tl Ao
+ AbA2 H2 + H2A2 Ao + AbA2(Wl + w2)Ao)'f} .
Here the E's denote the nucleonic free energies related to the accompanying projection operators
(A or 'f}). At next-to-next-to-leading order (NNLO), v = 9 - 3N, one finds:
��-3N) = 'f} ( H5 + AbA2 H3 + H3A2 Ao + At A2 H2 + H2A2 Al
+ A�Al Hl + HlAl A3 + AbAl HlA2 Al + At A2 HlAl Ao
(3.252)
+ AbAlwA3 + A�AlwAo + AbAl H3Al Ao + AbAlH4 + H4Al AO)
Solving eqs. (3.241)-(3.248) recursively one finds the express ions for the operators Aa AI'f} in terms
of the HK,'s. Inserting these into eqs. (3.250), (3.251) and (3.252) and performing straightforward
algebraic manipulations, we obtain the potential as
V(6-3N) eff
V (8-3N)
(3.253)
eff
V (9-3N)
eff
(3.254)
(3.255)