The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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(Eq - Ep )A(p, q)
[GeV-2]
2
1.6
1.2
0.8
0.4
q [GeV]
2. Low-momentum effective theories for two nucleons
Figure 2.7: The function (Eq - Ep)A(p, q) for A =
p [GeV]
2
GeV.
our case given by the exchange of the light meson, and to represent the effects of the heavy
meson exchange by adding a string of local contact interactions to the Hamiltonian. Proceeding
in this way one needs to introduce a cut-off to remove all ultraviolet divergences caused by the
contact interactions. Since a precise choice of the regulating function is of no relevance for the
low-energy observables, we can regard the same cut-off function as in the projecting formalism
discussed above. Speaking more precisely, we take the sharp cut-off. Now both the Hamiltonian
within the effective theory approach and the one derived from the unitary transformation are
defined over the same range of small momenta. That is why we can obtain the values of the
coupling constants accompanying the contact terms from matching the effective Hamiltonian to
the unitarily transformed one. It is commonly believed that the coupling constants, scaled by
some effective mass parameter Ascale, should have the property of "naturalness", which me ans they
should be of the order of one. Only then this expansion makes sense and the power-counting is
self-consistent. The value of the scale Ascale is obviously closely related to the radius of convergence
of this expansion. Let us first consider the simpler case when the pionic degrees of freedom are
integrated out. In that case, the low-energy N N interaction can be described entirely in terms of
contact terms and one expects the scale Ascale to be of the order of the pion mass m-rr . Therefore, all
physical parameters which describe the N N phase shifts up to center of mass momenta of the order
m-rr , such as the scattering length a and the effective range re , are expected to scale like appropriate
inverse powers of m7f' This is, however, not the case in nuclear physics: the NN scattering length
in the lSo-channel takes an unnatural large value, a = (-23.714 ± 0.013) fm » l/m7f. The
physics of this phenomenon is well understood (there is a virtual bound state very near zero