The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

144 4. The two-nuc1eon system: numerical results
waves OPE does already a fairly good job, but the improvement in some of these phases due to
the NNLO TPEP clearly underlines the importance of chiral symmetry in a precise description of
low-energy nuclear physics.
The most interesting situation in the peripheral partial waves appears for the 3 G5 phase,u In
these channel the OPE and the leading TPE are not sufficient to describe the phase shift correctly
at energies higher than 100 MeV and the largest discrepancy of our NLO predictions with the
data is observed. Adding the subleading TPE exchange (NNLO potential) that incorporates the
leading effects of the heavier meson exchanges and intermediate ..6.-excitations, which is hidden in
the values of the couplings Cl,3,4 allows to improve the NLO result considerably and to obtain an
excellent parameter free and cut-off independent description of the phase shift. Performing the
calculations with the NNLO-Ll potential we were even able to separate the leading effects of the
..6.'s. We will comment on that later on. It is comforting to have such a clear indication that the
values for Cl,3,4 fixed from 7f N scattering are consistent with N N calculations.
4.4 Deuteron properties
We now turn to the bound state properties. At NNLO (NLO), we consider the exponential
regulator (4.13) with n = 2 and A = 1.05 (0.60) GeV, which reproduces the deuteron binding
energy within an accuracy of about one third of a permille (2.5 percent).We make no attempt to
reproduce this number with better precision.12 The results for the phase shifts, which correspond
to these values of the exponential regulator, are very similar to those obtained with the sharp
cutoff A = 0.875 (0.50) GeV. For completeness, we list in table 4.6 the values of the coupling
constants in the 3 SI - 3 Dl channel corresponding to the exponential regulator.
NLO -0.0363 0.186 -0.190
NNLO -14.497 15.588 -4.358
NNLO-..6. -8.637 7.264 -0.447
Table 4.6: The values of the LECs as determined from the 3 SI _ 3 Dl channel. We use an
exponential cut-off with A = 0.6 GeV and 1.05 GeV at NLO and NNLO (NNLO-..6.), respectively.
The LEC 0 3 5 1 is in 104 GeV-2 , while the others are in 104 GeV-4 . The parameters of the NLO,
NNLO and NNLO-..6. potentials are obtained from fitting to the Nijmegen PSA.
In table 4.7 we collect the deuteron properties in comparison to the data and two realistic potential
model predictions (the pertinent formulae are given in app. H). We give the results for NLO and
NNLO. We note that the deviation of our prediction for the quadrupole moment compared to
the empirical value is slightly larger than for the realistic potentials. The asymptotic D / S ratio,
called ry, and the strength of the asymptotic wave function, As, are weIl described. The D-state
probability, which is not an observable, is most sensitive to small variations in the cut-off. At
NLO, it is comparable and at NNLO somewhat larger than the one obtained in the CD-Bonn or
ll Note that the 3Zj=I+I-phases show, in general, the largest effects when going from NLO to NNLO and thus are
most sensitive to the short range physics. The three extreme cases are the 1 P 2 -, 3 D 3 - and 3G5-waves. We assurne
some sort of cancelation in the potential in these channels, wh ich leads to a suppression of the OPE and the leading
TPE contributions.
12Note that the deuteron binding energy is not used to fit the free parameters in the potential.