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