The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.3. Going to higher energies: a toy model 45
exactly reproduced.21 For example, at NLO one can fix the constants Co and C 2 to reproduce
the scattering length and the effective range corresponding to the original potential (2.112). As
shown in table 2.4, both methods give similar results for the coupling constants.
Let us now make two observations concerning the obtained results. First, one observes a much
better convergence for the low-energy phase shift with the couplings from table 2.4 than with the
true values of Ci ' So This has to be expected, since now their values are optimized to obtain a good
description for the phase shifts at low-energies or for the effective range parameters and not to
precisely reproduce the true potential for low momenta, which is, in principle, not an observable.
The same holds also for the two-body binding energy given in table 2.5, which now shows a
much better convergence than with the true values of couplings. This indicates that the "fine
tuning" of the coupling constants may significantly improve the convergence for such "unnatural"
quantities22 like the small binding energy (or, equivalently, large scattering length). Indeed, small
changes in the values of the Ci's lead to large variations of the binding energy E, as follows from
the estimation (2.123), and may allow for E to take its correct value already at 'leading orders.
Secondly, one observes that all Ci'S (with the exception of C�) are within 25% of their exact values.
The deviation of the fitted value for C� from the exact one reminds us of the fact that such fine
tuning can produce sizeable uncertainties in higher orders in such type of cut-off schemes and thus
the interpretation of such values has to be taken with so me caution. This is, in fact, even a more
deep problem since the potential itself is not an observable. Unitary transformations pro duces
phase equivalent potentials without changing the observables. Although we have found only one
solution for the Ci 's shown in table 2.4, fixing the coupling constants from a fit to phase shifts or,
equivalently, to the first terms in the effective range expansion may, in general, produce multiple
solutions. An example of such multiple solutions was given in section 2.2. The precise connection
between the unitarily transformed and effective Hamiltonian is still to be clarified. This might
be quite important for the comparison between various transformed realistic N N interactions
acting only below a certain moment um cut-off and the effective potential derived using chiral
perturbation theory.
One can also perform a similar analysis for the two-nucleon 1 So channel, which is expected to
be even more troublesome than the 3 SI channel for the effective field theory approach since, as
already stated before, the scattering length takes an unnatural large value. We will not consider
here this case but note that all conclusions drawn above hold also in this particular channel. For
more details the interesting reader is referred to reference [124].
To end this section we would like to discuss one furt her issue. In section 2.2 we have considered an
effective theory for two nucleons at very low energies. In such a case even the longest range part
of the nuclear force can be considered as a short range effect. Thus, the effective potential entirely
consists of the contact interactions with increasing number of derivatives. It was demonstrated
that such an effective theory reproduces the effective range expansion but cannot go beyond it.
More precisely, only those first coefficients in the effective range expansion are reproduced which
have been used to fix the free parameters of the potential. No predictions for furt her coefficients
could be made. In our current model we have explicitly incorporated the long range part of the
underlying interaction in the effective Hamiltonian. As a consequence, the predictive power of the
effective theory increases. In table 2.5 we show the values of the binding energy and effective range
21 At NNLO we obtain elose but not exact values of the first four coefficients in the effective range expansion.
22 "Unnatural" means in this context that the corresponding observable takes a very small or very large value as
the result of a cancelation between some large numbers. For instance, the scattering length given in eq. (2.55) is
natural for all values of the coupling constant 9 not very elose to one and takes an unnaturally large value if 9 is
accidentally elose to one.