The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.3. Going to high er energies: a toy model 23
To estimate the quantum averages of the various terms in the expansion (2.5) they used the leading
order wave function of the low-energy bound state. (In the real world no bound state exists in this
partial wave.) Such a wave function corresponds to the potential V(p' ,p) = Co()(A2_p'2)e(A 2 _p2)
and is proportional to
'IjJ (O)(p) cx ()(A2 -p2)
mB+p2 ,
(2.58)
where B denotes the binding energy. For this case it has been shown that the quantum averages of
all terms in the expansion (2.5) with two and more powers of momenta are of the same size. Here
we refrain from doing similar estimations. We do not want to restrict ourselves to cut-off values
much smaller than 1/r as it is guaranteed by the second equality in (2.57). Requiring only aA » 1
does not lead to simple scaling properties of an coupling constants, which would be present for
rA « 1, as it follows from eqs. (2.45), (2.46). Also the leading order wave function of the bound
state might be a too naive approximation for calculating quantum averages. Rather , one should
take low-energy scattering states. A more careful investigation of convergence of the expansion
(2.5) would necessarily require a more complicated algebra and applying numerical methods. l3 In such a case the outcome might be different. We will co me back to this question in the next
section, where a more interesting example will be considered and the outcome is indeed different.
To summarize, we have illustrated the ideas discussed in sec. 2.1 with the example of two-nucleon
scattering at low energies. The effective potential consists of contact interactions with any number
of derivatives. Such an effective theory works as wen as the effective range expansion for the
inverse of the T -matrix. Cut-off regularization is an appropriate tool to provide finiteness of the
amplitude. One should keep the cut-off finite and of the order of the pion mass. There exists
an upper bound for the cut-off, above which no real solutions for the constants Ci are available.
Dimensional regularization with the standard subtraction scheme fails to provide an adequate
description of the amplitude if the scattering length is very large.
2.3 Going to higher energies: a toy model
In the last section we have considered the effective theory for nucleon-nucleon scattering at very
low energies. This analysis was quite general and applicable, in principle, to any underlying
theory. The only information we used is that the underlying interaction has a finite range of
order rv Such a theory can clearly fit the data for nucleonic three-momenta sm aller
1/M7r•
than M7r• It was shown that the effective theory in this case reproduces the effective range
expansion for the amplitude but cannot go beyond it. We will now consider another example
and go to higher energies. Our starting point is a simple model of the S-wave nucleon-nucleon
interaction, which contains long and short range parts. The parameters of this model are adjusted
to reproduce phase shifts in the 1 So and 3 SI channels. This model is our "fundamental" theory.
We will construct an effective theory for that case and keep explicitly the long range part of
the interaction. This is different to the considerations of the last section. Short range physics
is again represented by contact interactions. To make the scattering amplitude finite we will
introduce a sharp cut-off A in the momentum space, which is chosen between the two scales
corresponding to long and short range parts of the underlying force. Thus, our effective theory
is defined on the subspace of momenta below A. An interesting and new point is that we are
ahle to explicitly integrate out high moment um components from the underlying theory [124].
For that we divide the moment um space into two subspaces, spanning the values from zero to A
1 3 For instance, at higher orders in the derivative expansion one has more complicated non-linear conditions for
fixing the constants Ci .