The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

82 3. The derivation of nuclear forces from chiral Lagrangians
Because of the property eq. (3.191) of the operator A each single equation in eqs. (3.205) can be
expressed as
00 00
). iHrfJ + L ). iHj). j AT] + ). iHo). iAT] - ). iAT]HjT] - ). iAT]HoT] - L ). iAT]Hj). j AT] = 0
j=l j=l
(3.206)
The system of eqs. (3.205) is, clearly, too complicated to be solved exactly. In what follows, we will
apply the usual philosophy of effective theories. We are interested only in low-energy processes.
Therefore, we will expand the matrix elements of an effective potential in powers of the small
moment um scale Q, as it was proposed by Weinberg [73]. To calculate the operator A from the
system of coupled equations (3.205), we will again make use of the expansion in powers of Q. In
particular, the matrix elements of A will be classified by powers of Q by use of simple dimensional
analysis:
(3.207)
Here Q is again the mass scale corresponding to three-momenta of nucleons and four-momenta
of pions. The effective Hamiltonian acting on the purely nucleonic Fock space is defined in
eq. (3.197) and can be found from the original one if the operator A is known. It will be clear
from the power counting arguments that the matrix elements (3.207) with larger 1I A lead to an
effective potential suppressed by additional powers of Q. At first sight, this appears to be a
miracle, since (�I and 1