The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2. Secondly, we considered the realistic case of the nuclear interaction and presented a novel
approach, the projection formalism, to the problem of deriving the forces between few (two,
three, ... ) nucleons from effective chiral Lagrangians. For the case at hand, we first had to
modify the power counting rules proposed originally by Weinberg [73], since in the projection
formalism one decomposes the Fock space of nucleons and pions into subspaces with definite
nucleon and pion numbers. While in old-fashioned time-ordered perturbation theory the
resulting wave functions are only orthonormal to a certain order in the chiral expansion,
this problem does not occur in the projection formalism. Furthermore, in the previous
calculations based on time-ordered perturbation theory, the two-nucleon potentials turn
out to be energy-dependent, which is a severe complication for applying these in systems
with three or more nucleons (although to leading order, these recoil corrections are cancelled
by certain N-body interactions). We now summarize the results obtained in chapter 3.
• We start from the most general chiral invariant Hamiltonian for nonrelativistic nucleons
and pions and divide the full Fock space into the two subspaces T} and )... The first one
contains only purely nucleonic states, whereas the second one involves all remaining
states with nucleons and pions. To obtain an effective Hamiltonian acting on the T}space
we perform an appropriate unitary transformation, parametrized in terms of
the operator )"AT} via eq. (3.190). After projecting onto the states )..i with i pions,
the decoupling equation (3.192) turns into an infinite system of coupled equations
(3.205). We have proved that this system of equations can be solved perturbatively in
powers of the small momentum scale Q. For that we have developed an appropriate
power counting rule and analyzed the power of Q for all terms in eq. (3.206). We
have worked out the recursive prescription for solving the system of equations (3.205),
which allows to determine the operators )..i AT} for any finite i to any required order
in the Q-expansion. The effective Hamiltonian Heff acting on the T}-subspace can be
obtained via eq. (3.197) using the power counting rules (3.223)-(3.229).
• We have applied the projection formalism and have given the explicit expressions of
the two-nucleon potential to next-to-leading order (The lowest order potential comprises
the leading one-pion exchange and contact terms). In particular, we have also
calculated self-energy type corrections and one-loop corrections to these four-fermion
operators not considered before.
• We have discussed in detail the similarities and differences of the resulting potential
with those obtained from time-ordered perturbation theory. In particular, it turns
out that in our approach the isoscalar spin independent central and the isovector spin
dependent parts of the two-nucleon potential corresponding to the two-pion exchange
add up to zero. This was first noted in ref. [108] using a different scheme. Note that
in the energy dependent potential of the time-ordered perturbation theory there is no
such cancellation. However, the most salient features of the potential in the projection
formalism is its energy--independence and the orthonormality of the corresponding wave
functions.
• We have performed renormalization of the two-nucleon potential at NLO. In particular,
we have shown that all ultraviolet divergences can be removed by an appropriate
redefinition of the axial coupling constant gA and various couplings of the contact interactions.
The renormalized expressions for the N N potential at NLO agree with the
ones given by the Munich group which were obtained using the Feynmann diagram
technique.
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