The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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38<br />
(Eq - Ep )A(p, q)<br />
[GeV-2]<br />
2<br />
1.6<br />
1.2<br />
0.8<br />
0.4<br />
q [GeV]<br />
2. Low-momentum effective theories for two nucleons<br />
Figure 2.7: <strong>The</strong> function (Eq - Ep)A(p, q) for A =<br />
p [GeV]<br />
2<br />
GeV.<br />
our case given by the exchange of the light meson, and to represent the effects of the heavy<br />
meson exchange by add<strong>in</strong>g a str<strong>in</strong>g of local contact <strong>in</strong>teractions to the Hamiltonian. Proceed<strong>in</strong>g<br />
<strong>in</strong> this way one needs to <strong>in</strong>troduce a cut-off to remove all ultraviolet divergences caused by the<br />
contact <strong>in</strong>teractions. S<strong>in</strong>ce a precise choice of the regulat<strong>in</strong>g function is of no relevance for the<br />
low-energy observables, we can regard the same cut-off function as <strong>in</strong> the project<strong>in</strong>g formalism<br />
discussed above. Speak<strong>in</strong>g more precisely, we take the sharp cut-off. Now both the Hamiltonian<br />
with<strong>in</strong> the effective theory approach and the one derived from the unitary transformation are<br />
def<strong>in</strong>ed over the same range of small momenta. That is why we can obta<strong>in</strong> the values of the<br />
coupl<strong>in</strong>g constants accompany<strong>in</strong>g the contact terms from match<strong>in</strong>g the effective Hamiltonian to<br />
the unitarily transformed one. It is commonly believed that the coupl<strong>in</strong>g constants, scaled by<br />
some effective mass parameter Ascale, should have the property of "naturalness", which me ans they<br />
should be of the order of one. Only then this expansion makes sense and the power-count<strong>in</strong>g is<br />
self-consistent. <strong>The</strong> value of the scale Ascale is obviously closely related to the radius of convergence<br />
of this expansion. Let us first consider the simpler case when the pionic degrees of freedom are<br />
<strong>in</strong>tegrated out. In that case, the low-energy N N <strong>in</strong>teraction can be described entirely <strong>in</strong> terms of<br />
contact terms and one expects the scale Ascale to be of the order of the pion mass m-rr . <strong>The</strong>refore, all<br />
physical parameters which describe the N N phase shifts up to center of mass momenta of the order<br />
m-rr , such as the scatter<strong>in</strong>g length a and the effective range re , are expected to scale like appropriate<br />
<strong>in</strong>verse powers of m7f' This is, however, not the case <strong>in</strong> nuclear physics: the NN scatter<strong>in</strong>g length<br />
<strong>in</strong> the lSo-channel takes an unnatural large value, a = (-23.714 ± 0.013) fm » l/m7f. <strong>The</strong><br />
physics of this phenomenon is well understood (there is a virtual bound state very near zero