The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

80 3. The derivation of nuclear forces from chiral Lagrangians
The operator A has only mixed non-vanishing matrix elements:
A = >'A1] . (3.191)
The requirement for ut HU no longer to couple the two subspaces leads to the following nonlinear
equation for A:
>. (H - [A, H] - AHA) 1] = 0 , (3.192)
which is sometimes referred to as decoupling equation. Once the two spaces are decoupled one
can choose the Icp) component to be zero and the nucleonic states are orthonormal:
(3.193)
In the last step we have used the fact that Icp) = o.
In the case when the interaction Hamiltonian Hf can be treated as a small perturbation, it is
possible to solve Eq. (3.192) perturbatively to any given order. For instance, for the Hamiltonian
H represented by
00
H=Ho + LHn n==l
(3.194)
with the index n denoting the power of the coupling constant, one assurnes the operator A to be
of the form
00
(3.195)
The solution of eq. (3.192) to order n is then given by
(3.196)
Here, we denote the free-particle energy of the state 11]) by [. One can see from eq. (3.196) that
it is possible to find An for every n recursively, starting from Al.
As soon as the operator A is known, one can obtain the effective Hamiltonian, which operates
solely in the subspace Ix), via
(3.197)
as it follows from eqs. (3.189), (3.190). Expanding (1 + AtA)-1/2 and using eqs. (3.194), (3.195),
(3.196) one can obtain the effective Hamiltonian to any order in the coupling constant.
Several modifications are necessary by applying the formalism described above to effective Lagrangians
(Hamiltonians) and in particular to chiral invariant Lagrangians. First, the expansion
in powers of a coupling constant must be replaced by the expansion in powers of small momenta.
For doing that, power counting rules are necessary. Furthermore, one expects the operator >'A1]
to consist of an infinite number of terms to any order of Q caused by infinite number of vertices
in the Hamiltonian. We now show how these problems can be solved.