The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

188 F. The complete set of the N N contact interactions with two derivatives
For the terms without derivatives entering eqs. (F.7) and (F.12) one obtains:
G�O = -la12 '
(F.18)
Gs = -2Gs, (F.19)
Let us now take a closer look at the terms in eq. (F.ll). All interactions entering this equation
can be divided into three different classes: the terms al, 2 , 3 ,4 do not contain Sft, the terms a5,6,7,8
involve just one and the terms ag, ... ,18 two insertions of the spin-operator. One can make use of
partial integration for each of these classes separately, in order to eliminate redundant terms. For
example, the acterm can be expressed as a linear combination of the al, 2 , 3-interactions and, thus,
can be completely omitted. It is a matter of choice which terms are required as basis of independent
operators. For instance, to end up with the Lagrangian (F.13) one has to drop the a4,8,17,18-terms,
see eq. (F.18). We, however, point out that not all remaining interactions with two insertions of
the spin-operator (ag, ... ,16-terms in eq. (F.ll) or G7, ... ,lcterms in eq. (F.ll)) are independent.
Only 7 and not 8 such interactions are independent from each other, since one can express e.g. the
terms a9 ,lO,ll as linear combinations of the other a12 , ... ,16-couplings. To keep only independent
contact interactions in the effective Lagrangian we will use an operator basis, which is different
from eq. (F.13). More precisely, we choose the a = {I, 2, 3, 5, 6, 7, 12, 13, 14, 15, 16, 17, 18} -terms
(altogether 13 interactions instead of 14 entering eq. (F.13)).
Let us now work out the consequences of the requirement of these terms to be reparametrization
invariant. Performing the (infinitesimal) transformation (F.4) and replacing w by v at the end, g
we obtain the following constraints on the ai 's:
al - a2 + 2a3 1
'2a5 + a6
1
-a5 - a7 2
a14 + a16 - a18
1 1
a12 + -a15 - -a17
2 2
1 1
a1 3 + -a15 - -a17
2 2
0,
0,
0, (F.20)
0,
Consequently, only seven independent coupling constants enter the corresponding Lagrangian.
Denoting such couplings by Cl, ... ,7 and switching to the nucleon rest-frame system, our final
result for the reparametrization invariant set of independent N N contact interactions at order
L:1i = 2 reads:
.c��
- 1Cl [(NtVN)2 + (VNtN)2] - (Cl + C2)(NtVN) . (VNtN)
- 1C2(NtN) [NtV2N + V2NtN]
9 The complete effect of the reparametrization transformation (F.3) is given by the shift (F.4) of the nucleon
field, if no I/rn corrections are taken into account. Therefore, reparametrization invariance in that case turns into
Galilean invariance.
0,
0.