The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

78 3. The derivation o{ nuclear {orces {rom chiral Lagrangians
scale as l/Q. Let us now consider the two different cases shown in fig. 3.3. In the two-nucleon
center-of-mass system the energy denominator in the graph a) in fig. 3.3 is given by
1 1
Ei - E p2 Im -pl2 Im -vi q2 + M; (3.179)
where Ei is the initial energy of two nucleons and q = p' -p and p = 1P1, p' = Ip'l, q = 1iJ1. This
estimation is valid modulo 11m corrections. Consider now the second diagram. For the energy
denominator corresponding to the energy cut in fig. 3.3 b) we have:
1 1 m m .
Ei - E p2 Im -[2 Im + iE [2 (3.180)
p2 _
+ iE' rv Q2
Thus, the energy denominator in the second case is mlQ times larger than for the first diagram.
In the limit m ---+ (X) it becomes even infinitely large. Therefore, the power counting (3.177) is
not valid any more. To solve the problem with the large energy denominators arising from the
intermediate states with only two nucleons, Weinberg proposed to apply the technique of CHPT
not directly to the scattering amplitude but to the effective potential. The latter is defined as a sum
of all irreducible diagrams, i. e. those diagrams without pure nucleonic intermediate states. The
S-matrix elements are obtained by putting this potential into a Lippmann-Schwinger equation.
In fact, many equivalent schemes for deriving the effective potentials in nuclear and many-body
physics are known like that due to Bloch and Horowitz [183] or the Tainm-Dancoff approximation
[184]. The effective potential, derived using old-fashioned time-ordered perturbation theory,
possesses one unpleasant property: it is, in general, explicitly dependent on the energy of the
incoming nucleons and, as a consequence of this, not hermitian. Furthermore, the nucleonic wave
functions are not orthonormal in this approach [185].
One can avoid these problems by using the method of unitary transformation. It was already
applied successfully in cases where one has an expansion in a coupling constant, such as the
pion-nucleon coupling, see e.g. refs. [186]. In the following sections we will show how to apply
this method to the case of chiral perturbation theory for pions and nucleons, in which the small
momenta of extern al particles play the role of the expansion parameter. In fact, our considerations
are more general since they can be applied to any effective field theory of Goldstone bosons coupled
to some massive matter fields.
3.5 Bloch-Horowitz scheme and the method of unitary transfor­
mation
The method of unitary transformation (projection formalism) was already applied in the chapter 2
to decouple the spaces of small and large momenta in the quantum mechanical two-body system.
Here we would like to repeat the main points, mostly to establish our notation and keep the
section self-contained. We will also compare it with the Bloch-Horowitz scheme of deriving the
effective interactions.
A system of an arbitrary number of interacting pions and nucleons can be completely described
by a Schrödinger equation
(3.181)
with Ho (Hf) denoting the free (interaction) part of the Hamiltonian. In order to solve this
equation for nucleon-nucleon scattering, it is advantageous to project it onto a subspace
{IN), INN), INNN),
I