The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

4.4. Deuteron properties 145
the Nijmegen-93 potential. This increased value of PD is related to the strong NNLO TPEP. At
N3LO, we expect this to be compensated by dimension four counterterms. Altogether, we find
a much improved description of the deuteron as compared to ref. [78], where the binding energy,
magnetic moment and quadrupole moment were used in fits. Our results are almost as precise as
the ones obtained in the much more complicated and less systematic meson-exchange models.
11 Ed [MeV]
Qd [fm2]
77
rd [fm]
As [fm-l/2]
PD[%]
NLO 1 NNLO 1 NNLO-Ll 11 Nijm93 1 CD-Bonn 11
-2.1650 -2.2238 -2.1849 -2.224575 -2.224575
0.266 0.262 0.268 0.271 0.270
0.0248 0.0245 0.0247 0.0252 0.0255
1.975 1.967 1.970 1.968 1.966
0.866 0.884 0.873 0.8845 0.8845
3.62 6.11 5.00 5.76 4.83
Exp.
-2.224575(9)
0.2859(3)
0.0256(4)
1.9671(6)
0.8846(16)
-
Table 4.7: Deuteron properties derived from our chiral potential compared to two "realistic"
potentials (Nijmegen-93 and CD-Bonn) and the data. Here, rd is the root-mean-square matter
radius. An exponential regulator with A = 600 MeV and A = 1.05 Ge V at NLO and NNLO
(NNLO-Ll), in order, is used.
It is also interesting to compare our findings with the results reported by Park et al. [105]. There,
np scattering in the 3 SI-3 Dl channel as well as various deuteron properties are investigated. The
potential considered in this work consists of the leading OPE plus contact interactions without
and with two derivatives. To fix three free parameters corresponding to these contact terms the
authors of [105] use the deuteron binding energy and the experimental values for As and 77. For
the quadrupole moment they obtain values from 0.261 to 0.274 fm2 depending on the choice of
the cut-off. The D-state prob ability varies from 3.16% to 5.39%.
Kaplan et al. have recently reported on the NLO calculation of the deuteron charge radius and
the quadrupole moment in the KSW scheme [92]. They found analytic expressions for these
quantities. Having fixed three free parameters from the 3 SI phase shift, deuteron binding energy
and magnetic moment, they made the following predictions for the deuteron charge radius and
the quadrupole moment: reh = 1.89 fm 13 and Qd = 0.40 fm2. The large discrepancy for the
quadrupole moment is because this quantity is zero at LO in this formalism. Thus, the (formally)
NLO correction for Q d yields the first non-vanishing contribution.
The coordinate space S- and D-state wave functions obtained in our approach are shown in
fig. 4.14. At NLO they look qualitatively quite similar to the ones obtained from various potential
models. At NNLO one obtains a lot of structure in the wave functions below 2 fm. This is because
two additional spurious (unphysical) very deeply bound states appear in the 3S1 _3 Dl channel.
The binding energy of these states varies strongly by changing the cut-off. For the exponential
regulator with A = 1.05 GeV we get binding energies of EI = 47.1 GeV and E 2
= 2.5 GeV,
respectively. This values correspond to center-of-mass momenta of about a few Ge V which is
clearly out of the applicability range of the low-momentum effective theory. Furthermore, because
of such huge values of the binding energy these unphysical states obviously do not influence physics
in the energy region below 350 MeV that we are interested in and can, in principle, be integrated
1 3 The experimental value of the charge radius is 2.1303±O.0066 fm [212].