The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

22 2. Low-momentum effective theories for two nucleons
effective theory the potential is represented in form of an expansion in powers of momenta. Thus,
loop contributions can be expressed in terms of the power law divergences In and the integral
I, which yields the imaginary part of the S-matrix. For cut-off regularization these power law
divergences In are given by eq. (2.13) and are, in general, proportional to A n . It is crucial, that all
power law divergences vanish if one uses dimensional regularization, see eq. (2.49). That is why
the cancelation between tree- and loop-contributions to the amplitude, which is responsible for
large scattering length, becomes impossible in this case. This explains the failure of dimensional
regularization of the L8 equation for a ---+ 00. 12
Let us now make one more remark concerning the predictive power of our effective theory. The
question is, whether we are doing better than the effective range expansion (2.21). With other
words, once we have fixed free parameters from the first two terms of the effective range expansion,
can we make predictions for higher terms in this expansion? To answer this quest ion we can simply
calculate the next coefficient V2 in eq. (2.21) within the cut-off approach (2.34). We obtain from
eq. (2.40):
(2.56)
In the last equation we made use of the limit of a large scattering length. Thus, the value of
V2 obtained from the effective theory at next-to-leading order depends strongly on the cut-off
A. Clearly, the coefficient V2 cannot be predicted within our effective theory. This is because we
have not incorporated any additional information about the nucleon-nucleon interaction apart
from its parity and rotational invariance as weIl as symmetry under time revers al operation. Our
effective theory should equally weIl describe low-energy properties of any underlying theory, which
satisfy these constraints. For instance, the experimental value of the shape parameter V2 for the
neutron-proton 1 So channel is V2 rv -0.48 fm3. However, if the nucleon-nucleon interaction would
be as simple as the one in eq. (2.53), we would have V2 = O. Various symmetries that we have
incorporated in the effective theory do not allow to extract additional information about higherorder
terms in the effective range expansion from its first two coefficients. In some sense, the
effective theory considered in this section is trivial, since it only reparametrizes the effective range
expansion. Effective theory becomes non-trivial if one can incorporate more information from
the underlying theory. In the next section we will consider such an example. In chapters 3 and
4 we will show how to systematically incorporate the property of chiral symmetry into the N N
effective theory and demonstrate its remarkable predictive power.
Before finis hing this section we would like to comment on the convergence of the effective theory.
To show that the derivative expansion (2.5) of the effective potential converges one should demonstrate,
that the higher terms in this expansion give sm aller corrections to low-energy observables.
The authors of refs. [81], [83] have analyzed the convergence of the expansion (2.5) in the 1 S0 channel for cut-off values satisfying
1 4
- « A « - .
(2.57)
a Kr
12 Kaplan, Savage and Wise [90] pointed out that this is not a problem with dimensional regularization, but rather
with the subtraction scheme. Usually, only such poles of regularized expressions are subtracted, which correspond
to physical number of space-time dimensions d (i.e. d = dphys = 4). This allows to keep track of logarithmic
divergences. In the case of the LS equation with contact interactions the regularized expressions remain finite
since there are no logarithmic divergences at all. In order to account for linear divergences one can subtract poles
corresponding to d = dphys - 1 dimensions. Such power divergence subtraction scheme (PDS) looks, in some sense,
like a compromise between cut-off and dimensional regularizations: for n > 1 one has In --7 ° like in dimensional
regularization with the standard subtraction scheme, whereas h cx A, similar to cut-off approach.