The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

(j ± 1, 1jIVIJ =f 1, 1j)
Here, Pj (z) are the conventional Legendre polynomials. For j = 0 the two non�vanishing matrix
elements are
(OOOIV IOOO)
(1l01V 11l0)
211" [ 1 1 dz {Vc - 3Va + p ,2 p 2 (z 2 - l)VaL
211" [ 1 1 dz { ZVC + zVa + p ' p(z 2 - l)VSL
- ((p ,2 + p 2 )z _ 2p ' p) Vaq -
�
- q 2 Vaq - k
2 Va k} ,
+ p ,2 2
p z(1 - z2)VaL
((p ,2 + p 2 )z + 2p ' p) Va k} .
193
(G.3)
Note that sometimes another notation is used in which an additional overall "-" sign enters the
expressions for the off�diagonal matrix elements with l = j + 1, l' = j - 1 and l = j - 1, l' = j + 1.
Our results (G.2), (G.3) agree with the corresponding ones of ref. [193] apart from the off�diagonal
matrix elements (j ± 1, 1jIVaL a1 . (if x k) a2 . (if x k)1J =f 1, 1j) and with those ones for the on�
energy�shell matrix elements given in ref. [108] (up to an overall factor).
By deriving the effective N N potential we assume exact isospin invariance. In that case one can
express the operators Vcn a = {C, (T, SL, (TL, (Tq, (Tk} as
Therefore, the contribution to a state with total isospin I = 0, 1 is given by
(G.4)
(G.5)
The expressions (G.2), (G.3) can be brought into a different but equivalent form using the following
recurrence relation for the Legendre polynomials:
(G.6)