The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

5.2 Brief introduction into the KSW approach
Matching eq. (5.16) to eq. (5.17) yields the following scaling of the C's:
41f 1
C2n rv
mA A2n .
157
(5.18)
Consequently, the effective theory is perturbative: the leading order amplitude Ao = -Co is given
by the tree graph with Co at the vertix; Al = iCgmp/(41f) results from the one-Ioop diagram
with both Co-vertices and so forth.
In the case of a very large scattering length, lai » 1/ A, the expansion (5.17) has a very small
range of validity p < I/lai- Kaplan et al. proposed in that case to expand the amplitude in powers
of p/ A while retaining the factors ap to all orders:
A _ [ 1
_ 41f 1 r/2 2 (r/2)2 4 V 4
- 1+ P + P + 2 P + ... (5.19)
m l/a +ip l/a+ip (l/a +ip)2 l/a +ip
Therefore, for momenta p > I/lai the amplitude can be expressed as
00
A = L An ·
n=-l
(5.20)
As explained in ref. [91], using dimension regularization with the MS scheme does not lead to a
consistent effective theory for p > I/lai- This is also demonstrated in sec. 2.2. To deal with this
problem, the authors of [91] proposed to use dimensional regularization and the so-called power
divergence subtraction (PDS), i.e. to subtract the poles from the dimensionally regulated loop
integrals not only in the physical dimension, but also in the dimension one lower than the physical
one. This allows to take into account linear divergences. In a pionless effective theory one has to
deal with the (ultraviolet divergent) integrals of the form
(5.21)
where q == Igl and E = p2/m is the initial energy of the nucleons. Applying dimension regularization
with PDS leads tolO In --+ -(mEt (:) (ft+ip) . (5.22)
Chosing ft of the order of p, ft rv
couplings C 2n (ft):
p
» I/lai, leads to the following scaling properties of the
(5.23)
Further , each loop contributes a factor of p and the derivatives at vertices scale as p as weIl. With
these rules one can calculate the amplitude A to any order in the expansion (5.20). In particular,
the leading term A-I is given by sum of bubble diagrams with Co vertices; the next-to-Ieading
order term Ao results from dressing the C2-interaction to all orders by Co and so forth. Note
that since one uses a perturbative expansion for the amplitude A, the corresponding S-matrix is
unitary up to the calculated order. To find the phase shift one can use the relation
lOIn MS one would obtain the same result with /-l = O.
(5.24)