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The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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130 4. <strong>The</strong> two-nuc1eon system: numerical results<br />

1 E1ab [MeV] 11 NNLO* NNLO 1 Nijm PSA 1 VPI PSA 1 Nijm93 1 AV18 CD-Bonn<br />

h 147.735 147.727 147.747 147.781 147.768 147.749 147.748<br />

2 136.447 136.450 136.463 136.488 136.495 136.465 136.463<br />

3 128.763 128.781 128.784 128.788 128.826 128.786 128.783<br />

5* 118.150 118.196 118.178 118.129 118.240 118.182 118.175<br />

10* 102.56 102.67 102.61 102.41 102.72 102.62 102.60<br />

20 85.99 86.21 86.12 85.67 86.35 86.16 86.09<br />

30 75.84 76.14 76.06 75.46 76.40 76.12 75.99<br />

50* 62.35 62.79 62.77 62.12 63.36 62.89 62.63<br />

100* 42.33 43.06 43.23 42.98 44.33 43.18 42.93<br />

200 19.54 20.68 21.22 20.88 22.82 21.31 20.88<br />

300 4.15 5.58 6.60 5.08 8.44 7.55 6.70<br />

Table 4.2: 3 Sl np phase shift for the global fit at NNLO (sharp cut-off, A = 875 MeV) compared<br />

to phase shift analyses and modern potentials. <strong>The</strong> parameters of the NNLO potential are fixed<br />

by fitt<strong>in</strong>g the Nijmegen PSA at six energies (E1ab = 1,5, 10,25, 50(, 100) MeV). <strong>The</strong>se energies are<br />

marked by the star. <strong>The</strong> parameters of the NNLO* potential are chosen to reproduce exactly the<br />

scatter<strong>in</strong>g length and the effective range as described <strong>in</strong> the text.<br />

To show that our results are stable and do not depend on the fitt<strong>in</strong>g procedure we present <strong>in</strong><br />

table 4.5 the NLO* and NNLO* predictions. Here, the LECs are fixed directly from the first<br />

effective range parameters, as discussed above. Overall, the agreement between the numbers <strong>in</strong><br />

tables 4.4 and 4.5 is quite satisfactory. <strong>The</strong> NLO results turn out to be slightly more sensitive<br />

to the fitt<strong>in</strong>g procedure. As is po<strong>in</strong>ted out <strong>in</strong> ref. [208], one also can perform the moment um<br />

expansion for E1 <strong>in</strong> a similar way as for the S-waves. <strong>The</strong> authors of the reference [208] show the<br />

values of the correspond<strong>in</strong>g parameters extracted by themselves from the Nijmegen PSA. <strong>The</strong>y<br />

use, presumably, the different (Stapp) parametrization of the S-matrix <strong>in</strong> that case. We have not<br />

found any orig<strong>in</strong>al reference of the Nijmegen group concern<strong>in</strong>g the moment um expansion for E1 .<br />

We thus refra<strong>in</strong> from discuss<strong>in</strong>g this issue any further.3<br />

For completeness, we collect the experimental values for the S-wave scatter<strong>in</strong>g lengths and effective<br />

ranges:<br />

as = (-23.758 ± 0.010) fm , Ts = (2.75 ± 0.05) fm ,<br />

at = (5.424 ± 0.004) fm , Tt = (1.759 ± 0.005) fm .<br />

( 4.27)<br />

( 4.28)<br />

Quite recently the results for the NNLO calculation of the phase shifts <strong>in</strong> the S, P and D channels<br />

with<strong>in</strong> the KSW scheme [91] have been published [211]. Here we would like to compare our<br />

f<strong>in</strong>d<strong>in</strong>gs with those from ref. [211]. <strong>The</strong> lead<strong>in</strong>g and non-perturbative contribution to the N N Smatrix<br />

<strong>in</strong> S channels results <strong>in</strong> the KSW approach from summ<strong>in</strong>g up an <strong>in</strong>f<strong>in</strong>ite series of the contact<br />

3Note, however, that <strong>in</strong> order to fix the LEes <strong>in</strong> the 3S1 _3 D1 channel we have used the lead<strong>in</strong>g coefficient<br />

91 = 1.66 fm3 <strong>in</strong> the momentum expansion of the EI . This value agrees with the one given <strong>in</strong> ref. [208]. In pr<strong>in</strong>ciple,<br />

we could take apart from the 3 SI scatter<strong>in</strong>g length and effective range the deuteron b<strong>in</strong>d<strong>in</strong>g energy as the third<br />

quantity to fix the three free parameters <strong>in</strong> the potential. However we refra<strong>in</strong> from do<strong>in</strong>g that because of a strong<br />

correlation between the deuteron b<strong>in</strong>d<strong>in</strong>g energy and the 3 SI effective range parameters.

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