The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

6 1. Introduction
pion-nucleon system as well as for the processes with electroweak probes [71]. For arecent review
see ref. [72].
Motivated by successful applications of CHPT in the 7f7f and 7f N sectors Weinberg proposed in
1990 to extend the formalism to the N N interaction [73]. The crucial difference is, however,
that the nucleon-nucleon interaction is non-perturbative at low energies: it is strong enough to
bind two nucleons in the deuteron. Thus, direct application of CHPT to the N N amplitude will
necessarily fail. The way out of this problem, proposed by Weinberg, is to apply CHPT not to
the amplitude but to a kernel of the corresponding integral Lippmann-Schwinger (LS) equation.
Such a kernel (or potential) can be defined within time-ordered perturbation theory as a sum over
all irreducible diagrams with two incoming and outgoing nucleon lines. Irreducible means here
that no pure 2N intermediate states are allowed. Reducible diagrams are then generated due to
iterations of the kernel in the LS equation. Weinberg has shown that a systematic power counting
for the potential can be derived in a similar manner as in the case of the 7f7f and 7f N systems.
The corresponding Lagrangian should be extended to allow apart from the 7f7f and 7f N also N N
contact interactions, which are not constrained by chiral but only by isospin symmetry.
These ideas have been extensively studied by the Texas-Seattle group [74], [75], [76], [77], [78].
They have obtained an energy dependent two-nucleon potential at next-to-next-to-leading order.
With altogether 26 free parameters6 they were able to achieve a qualitative agreement with the
N N scattering data and some deuteron properties. Also three-body forces have been considered
by this group. Applying this formalism to many-body problems allows to establish a beautiful
hierarchy of the two- and many-body forces: the three-body forces should be weaker than the
two-body ones, the four-body forces should be weaker than the three-body ones and so on.
Since that time the derivation of the nucleon-nucleon interaction using the technique of effective
theories became a subject of quite remarkable interest and intense discussions. Cohen and collab
orators [79], [80], [81], [83] considered in detail an effective theory with pions integrated out.
In such a case simple analytic calculations for the two-nucleon S-matrix can be performed, since
all interactions between nucleons are of the contact type. They examined different regularization
schemes for divergent integrals in the LS equation like dimensional and cut-off regularizations and
made some interesting observations. First, it turned out that in such a pionless effective theory
the cut-off could be taken to infinity only if the effective range parameter of the interaction is
negative. This conclusion follows, in fact, from a theorem which had been proven by Wigner long
time aga [84] and is based only on such general principles like causality and unitarity. The theorem
says, that if a potential vanishes beyond some range R, then the rate d6(k)/dk at which the phase
shift can change with energy is bounded from below by some function of R, k and 6(k). Secondly,
an effective theory with a finite cut-off was found not to be "systematic" in the sense, that higherorder
terms in the potential do not get systematically smaller as the order is increased. Thus, the
expansion does not seem to converge. We will comment more on that in the chapter 2. Finally,
the use of cut-off schemes and dimensional regularization was shown to lead to different results
for the scattering amplitude. The scattering amplitude calculated with dimensional regularization
only maps to the effective range expansion for on-shell momenta k « 1/ JaTe, where a and Te
are the scattering length and the effective range. For such small momenta both schemes produce
identical results in agreement with the effective range expansion. The experimental values for the
S-wave np scattering lengths and effective ranges are
6 Some of these parameters are redundant.
as = (-23.758 ± 0.010) fm ,
at = (5.424 ± 0.004) fm ,
Ts = (2.75 ± 0.05) fm ,
Tt = (1.759 ± 0.005) fm .
(1.1 )
(1.2)