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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2.3. Go<strong>in</strong>g to higher energies: a toy model 41<br />
numerically, we can determ<strong>in</strong>e the constants Ci by fitt<strong>in</strong>g eq. (2.116) to V'(q', q) - VIight (q', q). This<br />
is done numerically us<strong>in</strong>g the standard FORTRAN subrout<strong>in</strong>es for polynomial fits to functions of<br />
one variable and tak<strong>in</strong>g the correspond<strong>in</strong>g polynomials typically of order ten to eleven. Once this is<br />
done for the functions V(q, 0) and V(q, q) all constants Ci, C: <strong>in</strong> eq. (2.117) can be easily evaluated.<br />
For the potential parameters used so far and the choices of A = 400 MeV and A = 300 Me V, the<br />
result<strong>in</strong>g constants are given <strong>in</strong> table 2.1:<br />
Co C' 4<br />
6<br />
1 A = 400 MeV 11 11.23 1 -32.27 1 86.73 1 113.9 -913.6<br />
I A = 300<br />
-265.2<br />
MeV 11 11.15 I -32.93 I 85.76 I 115.6 -232.6<br />
C'<br />
-783.7<br />
Table 2.1: <strong>The</strong> values of the coupl<strong>in</strong>g constants Ci, q <strong>in</strong> [GeV(-2-i )] for two choices of the cut-off<br />
A.<br />
Naturalness of the coupl<strong>in</strong>g constants Ci, q me ans that<br />
(2.118)<br />
C2n<br />
- = anAscale 2 ,<br />
C 2n+2<br />
where the an are numbers of order one. Indeed, as one can read off from table 2.1, such a common<br />
scale exists, namely<br />
Ascale =<br />
600<br />
MeV . (2.119)<br />
This is a reasonable value <strong>in</strong> the sense that it is very elose to the mass /1H of the heavy mesons,<br />
which is <strong>in</strong>tegrated out from the theory. Stated differently, the value for Ascale agrees with naive<br />
expectations. As it turns out, even for cut-off values like A = 300 MeV, there is only a small<br />
difference between the heavy meson mass and the ensu<strong>in</strong>g natural mass scale.<br />
Another observation is that the values of the Ci, CI depend very weakly on the concrete choice<br />
of the cut-off A. Clearly, for such an expansion of the heavy mass exchange <strong>in</strong> terms of contact<br />
<strong>in</strong>teractions to make sense, A has to be chosen below /1H. Furthermore, s<strong>in</strong>ce we explicitly keep<br />
the light meson, A should not be smaller than the mass /1L. If one were to select such a value<br />
for the cut-off, one could also consider the possibility of expand<strong>in</strong>g the light meson exchange <strong>in</strong> a<br />
str<strong>in</strong>g of contact terms. We do, however, not pursue this option <strong>in</strong> here. <strong>The</strong>refore, we conelude<br />
that one should set A < Ascale but it is not possible to f<strong>in</strong>d a more precise relation. In fig. 2.10<br />
MeV.<br />
we show how weIl the potential V' (q', q) is reproduced by the truncated expansion eq. (2.116) for<br />
A = 300<br />
We have also calculated the two-body b<strong>in</strong>d<strong>in</strong>g energy and the phase shifts us<strong>in</strong>g the form<br />
eq. (2.116) with the constants given <strong>in</strong> table 2.1. <strong>The</strong> correspond<strong>in</strong>g results are shown <strong>in</strong> fig. 2.11<br />
and <strong>in</strong> table 2.2. <strong>The</strong> agreement with the exact values is good. However, one sees that terms<br />
of rat her high order should be kept <strong>in</strong> the effective potential Vcontact <strong>in</strong> order to have the b<strong>in</strong>d<strong>in</strong>g<br />
energy correct with<strong>in</strong> a few percent. Note, however, that the value of the b<strong>in</strong>d<strong>in</strong>g energy is<br />
unnaturally small compared to a typical hadronic scale like the pion mass or the scale of chiral<br />
symmetry break<strong>in</strong>g. <strong>The</strong> phase shifts are described fairly weIl for k<strong>in</strong>etic energies (<strong>in</strong> the lab) up to<br />
about 100 MeV if one reta<strong>in</strong>s the first three terms <strong>in</strong> the expansion eq. (2.116). For A = 400 MeV,<br />
one can not expect any reasonably fast convergence for the b<strong>in</strong>d<strong>in</strong>g energy any more. This can<br />
be traced back to the fact that one is elose to the radius of convergence for momenta elose to<br />
the cut-off. More specificaIly, with q = q' = 400 Me V, the pert<strong>in</strong>ent expansion parameter is<br />
(q,2 + q2)j A;cale ::::' 0.9. <strong>The</strong> ensu<strong>in</strong>g very slow convergence is exhibited <strong>in</strong> table 2.2.<br />
Although we have found that Ascale rv 600 MeV and, therefore, the expansion of V' <strong>in</strong> terms of<br />
contact terms seems to converge for the chosen value of the cut-off A = 300 MeV, the ultimative