The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

Thus, the effective theory with dimensional regularization has a quite small range of validity
and provides an extremely poor fit to the data for the ISO partial wave [85]. The problem with
the dimensional regularization is that it does not account for power law divergences of the type
Jooo dnq qm, which turn out to be important for the renormalization of the LS equation. In dimensional
regularization divergent integrals in four dimensions are extended to an arbitrary number
of dimensions D. After the regularization is performed, they are typically expressed in terms of
f-functions which depend on D. Ultraviolet divergences correspond to poles of these functions.
They have to be subtracted in order to provide a finite result. In the minimal subtraction scheme
(MS), which is frequently used in field-theoretical calculations with renormalizable theories, all
1/(D-4) poles are subtracted before taking the limit D --+ 4. This allows to remove all logarithmic
divergences. In the effective theory for N N scattering without pions no such logarithmic divergences
appear and the regularized integrals in four dimensions remain finite and scale independent.
The only divergences that appear in such an effective theory are the power law divergences, which
vanish after performing dimensional regularization. Kaplan, Savage and Wise (KSW) proposed to
subtract also poles in D = 3 dimensions to keep trace of power law divergences [90]. They called
this formalism power divergence subtraction (PDS). Using PDS, Kaplan et al. worked out a new
power counting scheme (KSW approach) for systems with a large scattering length and extended
it to take into account pionic degrees of freedom. Equivalent formalisms with different regularization
schemes are discussed in refs. [86], [87], [88]. The leading and non-perturbative contribution
to the scattering amplitude in these methods is due to N N contact interactions without derivatives.
Pion exchanges start to contribute at subleading order and can be treated perturbatively.
That is why analytic calculations are possible. The corresponding S-matrix is (approximately)
unitary7 and does not depend on the renormalization scale which is introduced due to PDS. The
expansion for the S-matrix is expected to break down at momenta k rv 300 MeV. For a series
of interesting applications of the KSW approach to various processes see references [90]-[99]. In
spite of a successful description of many observables at low energies the formalism fails badly to
describe the shape parameters in the ISO and 3S1 _3 D 1 channels, as has been shown by Cohen and
Hansen [100], [101]. They have pointed out that predictions for these quantities would provide
a sensitive test for the correct inelusion of pionic physics. Note furt her that different potential
models give very elose predictions for the shape parameters. This is, according to ref. [102] , due
to the correct treatment of the one-pion exchange. Thus, it remains unelear, whether such a
perturbative treatment of the pion exchange is justified8 [103].
An interesting work within the power counting scheme proposed by Weinberg has been performed
by Park et al. [104], [105]. They concentrated on np scattering for the ISO partial wave and the
deuteron channel and were able to reproduce the empirical scattering phase shifts up to momenta
k rv 300 Me V. The potential they considered contained apart from the contact N N interaction
without and with two derivatives also the leading OPE term. The cut-off dependence of the
results was investigated. Quite recently they also ineluded the leading two-pion exchange as weH
as contact interactions with four derivatives [106]. For the application of the effective field theory
technique to the solar proton burning process p + p --+ d + e+ + Ve see reference [107].
The Munich group considered recently the peripheral np partial waves within chiral perturbation
theory [108]. No contact terms without and with two derivatives contribute in such partial waves
(starting from the D-waves) because of the large values of angular momenta. The interaction is
7 A different approach has been recently proposed by Lutz [89], which preserves unitarity of the S-matrix exactly
and allows far the description of the N N phase shifts at high er energies.
8 Certainly, for very small energies E « M" � 140 Me V one can even completely integrate out pionic degrees of
freedom.
7