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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4.3. Phase shifts<br />
1 Elab [MeV] 11 NNLO*<br />
h 62.071<br />
2 64.472<br />
3 64.671<br />
5* 63.659<br />
10* 60.02<br />
20 53.66<br />
30 48.55<br />
50* 40.49<br />
100* 26.30<br />
200 7.63<br />
300 -6.41<br />
NNLO 1 Nijm PSA I VPI PSA 1 Nijm93 1 AV18 CD-Bonn I<br />
62.063 62.069 62.156 62.065 62.015 62.078<br />
64.469 64.573 64.573 64.460 64.388 64.478<br />
64.671 64.762 64.762 64.650 64.560 64.671<br />
63.663 63.708 63.708 63.619 63.503 63.645<br />
60.03 59.96 60.00 59.94 59.78 59.97<br />
53.68 53.57 53.77 53.54 53.31 53.56<br />
48.58 48.49 49.00 48.42 48.16 48.43<br />
40.54 40.54 41.66 40.38 40.09 40.37<br />
26.38 26.78 27.86 26.17 26.02 26.26<br />
7.76 8.94 7.86 7.07 8.00 8.14<br />
-6.24 -4.46 -5.55 -7.18 -4.54 -4.45<br />
Table 4.3: ISO np phase shift for the best 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 />
I SO NLO -23.555 2.64 -0.58 5.4 -31<br />
I SO NNLO -23.722 2.68 -0.61 5.1 -30<br />
ISO NPSA -23.739 2.68 -0.48 4.0 -20<br />
3S 1 NLO 5.434 1.711 0.075 0.77 -4.2<br />
3S 1 NNLO 5.424 1.741 0.046 0.67 -3.9<br />
3S 1 NPSA 5.420 1.753 0.040 0.67 -4.0<br />
Table 4.4: Scatter<strong>in</strong>g lengths and range parameters for the S-waves at NLO and NNLO (global<br />
fits) compared to the Nijmegen PSA (NPSA). <strong>The</strong> values for V2, 3 , 4 <strong>in</strong> the 1 So channel are based<br />
on the np Nijm II potential and the values of the scatter<strong>in</strong>g length and the effective range are<br />
taken from the ref. [209]. <strong>The</strong> effective range parameters for the 3S 1 _3 D 1 channel are discussed<br />
<strong>in</strong> [102].<br />
<strong>in</strong>teractions without derivatives. <strong>The</strong> first corrections are given by dress<strong>in</strong>g the one-pion exchange<br />
and the contact <strong>in</strong>teractions with two derivatives by the lead<strong>in</strong>g order amplitude. This is the<br />
crucial difference to our power count<strong>in</strong>g scheme, <strong>in</strong> which the OPE diagrams are of the same size as<br />
the lead<strong>in</strong>g contact <strong>in</strong>teractions and thus should both be treated non-perturbatively. <strong>The</strong> NNLO<br />
corrections <strong>in</strong> the KSW scheme are given by various diagrams <strong>in</strong>clud<strong>in</strong>g contact <strong>in</strong>teractions with 0,<br />
2 and 4 derivatives as weH as pion exchange graphs. For more details see ref. [211]. Because of the<br />
perturbative treatment of the pion exchanges, the authors of ref. [211] could perform an analytic<br />
calculation of the amplitude <strong>in</strong>clud<strong>in</strong>g its renormalization. <strong>The</strong> result<strong>in</strong>g S-matrix satisfies the<br />
perturbative unitarity condition <strong>in</strong> the sense of the KSW power count<strong>in</strong>g. Furthermore, at each<br />
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