The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

88 3. The derivation of nuclear forces from chiral Lagrangians
+ �(\ [(NtVN)2 + (VNtN)2] + ((\ + 02 )(NtVN) . (VNtN)
+ �02 (NtN) [NtV2N + V2NtN]
+i�63{ [(NtVN) . (VNt x iJN) + (VNtN) . (NtiJ x VN)]
+ (NtN)(VNt . iJ x VN) + (NtiJN) . (VNt x VN)} (3.237)
1 ( 1 - -
+ 2 2 C 4 (6ik6jl + 6il6kj) + C56ij6kl
x [(OiOjNt17kN) + (Nt17kOiOjN)] (Nt171N)
+ (�06 (6ik6jl + 6il6kj) + (05 - 67)6ij6k1) (Nt17kOiN)(ojNt171N)
1 ( 1 - -
-
+ 2 2 (C4 - C6) (6ik6jl + 6i/6kj) + C76ij6kl
x [(OiNt17kOjN) + (ojNt17kOiN)] (Nt171N) ,
1i5 �El (NtN) (NtrN) . (NtrN) + �E 2 (NtN) (NtriJN) . . (NtriJN)
+ �E3 [(NtriJN) x x (NtriJN)] .. (NtriJN) .
)
)
(3.238)
Here we have shown explicitly only the operators 1i". leading to non-vanishing TjH".Tj, ),1 H".),l,
),1 H".Tj, ),2 H".Tj, ),2 H".),l, ),4H".Tj and h. c., which we will need in our furt her calculations. The
corresponding index /'l, is defined in eq. (3.211). The operators with nucleon fields and three or
. . more pion fields are irrelevant. The symbols ' ' ' , x x
' me an that the appropriate products in
co-ordinate and isospin space have to be taken. Note that one has, in principle, four possible
contact terms (two more involving r) in eq. (3.234). However, they can be reduced to two after
performing anti-symmetrization of the potential. This will be explained below. Furt her , the
Hamiltonian used in this paper is always taken in normal ordering. Finally, we have used the
standard (in the one-nucleon sector) notation for the Cl , C3 and C4 terms, which is different from
the corresponding one given in the reference [78] 39 and also our El, 2 ,3 couplings differ from those
defined in [77]. In particular, EI = EU4, E2 = E!J./4 and E3 = Ej/8, where we denote by E* the
corresponding parameters from reference [77].
The Hamiltonian given in [78] contains apart from the terms enumerated above the two additional
interactions
(3.239)
which lead to significant contributions to the two-nucleon potential at next-to-leading order.
However, as argued in [127], no corresponding terms appear in the relativistic Lagrangian after an
appropriate nucleon field redefinition is performed. Therefore, terms of such type in the nonrelativistic
Lagrangian may only represent l/m-corrections, that are irrelevant for our calculations
because of the counting eq. (A.14).
39 In ref. [78] these are the EI-E3 terms.