The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

potential. Furt her , we will apply cut-off functions, which do not introduce any additional angular
dependence. That is why the contact interactions will only contribute to the S- and P-waves and
to the E1 mixing angle. The results for higher partial waves are parameter free predictions. That
establishes a connection with the recent work by the Munich group [108], [109]. Furthermore, we
will demonstrate how to redefine the contact interactions in order to have separate parameters
in each low partial wave. This allows for an enormous simplification of the fitting procedure.
FinaIly, our results for the S-waves and for various deuteron properties will be shown to achieve
the level of precision of modern phenomenological potentials. Thus, we believe, that quantitative
calculations for the N N system using the effective field theory approach are possible.
The manuscript is organized as follows. In chapter 2 we will give an introduction to effective
theories. For a better understanding of the philosophy of effective theories the quantum mechanical
two-body scattering problem will be considered. We will apply the method of unitary
transformation [51], [53] to a model two-nucleon Hamiltonian in order to construct from it an
effective potential, which operates in a subspace of momenta below a given cut-off A [123], [124].
For concrete numerical investigations we will restrict ourselves to a potential of the Malfiiet-Tjon
type [125], [126]. This force consists of two terms, an attractive one due to the exchange of a light
meson and a repulsive term parametrized in terms of a heavy meson exchange. The S-matrices
in the full space and in the subspace of low momenta after performing the unitary transformation
will be shown to be identical. Since we are interested only in the region of low momenta, weIl
below the mass of the heavy meson, we can "integrate out" the heavy meson. More precisely,
we will keep in the potential the light meson exchange unchanged and represent effects of the
heavy meson exchange in terms of the N N contact interactions. This i,s rat her similar to what
is usually done for the real two-nucleon system. The crucial advantage of our model is, however,
that the exact effective Hamiltonian for low momenta is known from the unitary transformation.
This allows us to compare these two effective theories, the one with contact forces and the exact
one, and to study effects of fine tuning which are important for reproducing the large value of the
scattering length. FinaIly, we will address some furt her issues related to the chiral perturbation
theory approach of the two-nucleon system.
We will begin chapter 3 with a brief introduction to chiral symmetry and construct the twoand
three-nucleon potential, based on the most general chiral effective pion-nucleon Lagrangian
using the method of unitary transformations [127]. For that, a power counting scheme consistent
with this projection formalism has been developed. The details are discussed in appendices A, B.
Using this power counting scheme it has been shown how to calculate the effective potential to
an arbitrary order in the small moment um scale Q. In appendix C we have also generalized the
above approach to the case of the so-called "small scale expansion" for the potential, in which
not only the pion mass and external momenta of the nucleons but also the tlN mass splitting
are considered as small quantities and are treated in the same footing. Such an expansion allows
to take into account effects in the potential due to intermediate excitations of the tl-isobar. To
the best of our knowledge, this is the first time that the method of unitary transformation is
systematically applied in the context of chiral effective field theory. We discuss in detail the
similarities and differences to the existing chiral nucleon-nucleon potentials and show that, to
leading order in the power counting, the three-nucleon forces vanish lending credit to the result
obtained by Weinberg using old-fashioned time-ordered perturbation theory. We will also consider
the problem of the renormalization of the potential. For that we will use the express ions for the
divergent integrals presented in appendix D and perform anti-symmetrization of various contact
interactions as explained in appendix E. Finally, we have analyzed the contact terms in the
effective Lagrangian with four nucleon legs and two derivatives and found that seven of fourteen
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