The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

156 5. Isospin violation in the two-nuc1eon system
Born amplitudes for the lowest order eIB and eSB operators between the various two-nucleon
states takes the form
_ (�:)
_ (�:)
a (E�l) + E�2))
a (E�l) _
E�2))
(5.13)
where we will determine the coupling constant E�1,2) later on and also derive the scaling properties
of the E�;,2). The terms with the superscript '(1)' refer to eIB whereas the second ones relevant
for eSB are denoted by the superscript '(2)'. Higher order operators are denoted accordingly.
There is, of course, also a eIB contribution to the np matrix element. To be consistent with the
charge symmetrie calculation of ref. [91J, we absorb its effect in the constant D2, i.e. it amounts
to a finite renormalization of D2 and is thus not observable. In eq. (5.13), p = JmEcms is the
nucleon cms moment um.
5.2 Brief introduction into the KSW approach
We will now give a very brief introduction into the KSW formalism and repeat so me basic points of
ref. [91J. It is weIl known that the scattering lenghts a in both np S-channels take unnatural large
values. Let us first consider a pionless theory. In such a case one can calculate the two-nucleon
T-matrix analytically, see sec. 2.2, since all interactions in the effective Lagrangian are of contact
type. As discussed in section 2.2, the range of applicability of such an effective theory is zero
in the case of infinitly large scattering length, if ordinary dimensional regularization is used. In
what follows we will adopt the notation of ref. [91J and work with the amplitude A instead of the
T-matrix defined in eq. (2.6). The connection between the amplitude and the on-shell T-matrix
Ton (p) is gi yen by
= A( ) _ Ton(p) p 27r2 '
The tree level amplitude in the S-channels can be expressed as
Atree =
00
- L C2np2n ,
n=O
(5.14)
(5.15)
where C2n are some constants. Using dimensional regularization with the minimal subtraction
= scheme (MS), which amounts to subtracting poles in the physical dimension (D 4), one finds
the amplitude A in the form:
(5.16)
Here we have used the fact that the power law divergent integrals (like those one in eq. (2.49))
vanish after performing dimensional regularization. If the scattering length would be of a natural
size, i.e. of the order a rv 1/ A, where A is ascale entering the values of the remaining effective range
and shape parameters, which is comparable with the pion mass, one could rewrite the effective
range expansion (2.21) for the inverse T-matrix into the expansion for the amplitude A:
A =
�
An =
00
- ---;
7ra . ar 2 2 P
4 [ ( )
;: 1 -
zap
( 3)]
+ 2 - a p + 0
A 3 . (5.17)