21.12.2012 Views

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

SHOW MORE
SHOW LESS

You also want an ePaper? Increase the reach of your titles

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

NNLO, this range is larger and extends from 650 to 1000 MeV. This can be understood<br />

from the chiral TPEP, which at NNLO <strong>in</strong>cludes 1f1f correlations. <strong>The</strong>se <strong>in</strong>troduce a<br />

new mass scale well above twice the pion mass.<br />

• We have shown that the NLO coupl<strong>in</strong>gs can be comb<strong>in</strong>ed <strong>in</strong> such a way that each<br />

comb<strong>in</strong>ation feeds <strong>in</strong>to one partial wave, see eqs. (4.16)-(4.23).1 More precisely, the<br />

n<strong>in</strong>e coupl<strong>in</strong>gs with four nucleon legs can be determ<strong>in</strong>ed uniquely by a fit to the two Swaves,<br />

four P-waves and the mix<strong>in</strong>g parameter E1 for nucleon laboratory energies below<br />

100 MeV. This simplifies the fitt<strong>in</strong>g procedure enormously as compared to ref. [78],<br />

where a global fit <strong>in</strong> all low partial waves has been performed. As expected from the<br />

power count<strong>in</strong>g underly<strong>in</strong>g the EFT, the fits improve when go<strong>in</strong>g from LO to NLO to<br />

NNLO, compare fig. 4.l.<br />

• At NNLO, the result<strong>in</strong>g S-waves are of very high precision (for nucleon laboratory<br />

energies below 300 MeV), see e.g. tables 4.2,4.3 and fig. 4.4. <strong>The</strong> so-called range<br />

parameters collected <strong>in</strong> table 4.4 agree with what is found <strong>in</strong> the phase shift analysis.<br />

<strong>The</strong> P-waves are mostly well described, <strong>in</strong> particular the mix<strong>in</strong>g parameter E1 is <strong>in</strong><br />

good agreement with the phase shift analysis. We also note that above nucleon cms<br />

momenta of about 150 MeV, our NLO and NNLO results are far better than the ones<br />

obta<strong>in</strong>ed <strong>in</strong> the KSW scheme at NLO and NNLO.<br />

• All other partial waves are free of parameters. <strong>The</strong> D-waves, <strong>in</strong> particular 3 D1 and 3 D 3<br />

are very well described. We have also discussed the cut-off sensitivity of these results.<br />

<strong>The</strong> NNLO TPEP is too strong <strong>in</strong> the trip let F-waves. For the peripheral waves, we<br />

recover the results of the Munich group [108], namely that <strong>in</strong> most cases OPE works<br />

well but chiral NNLO TPEP clearly improves the description of some partial waves like<br />

e.g. 3G5, 3 H5 or 3 h.<br />

• <strong>The</strong> deuteron properties are mostly well described, at NLO and NNLO, compare table<br />

4.7. At NNLO, the deuteron wave functions show some <strong>in</strong>terest<strong>in</strong>g structure due<br />

to the appearance of two very deeply bound states. <strong>The</strong>se are an artifact of the NNLO<br />

approximation. <strong>The</strong>y have no <strong>in</strong>fluence on low energy properties and can be projected<br />

out completely from the theory. Our precise deuteron wavefunctions can be used for<br />

pion photoproduction, pion-deuteron scatter<strong>in</strong>g or Compton scatter<strong>in</strong>g off deuterium<br />

(still, the hybrid approach proposed by We<strong>in</strong>berg [114] rema<strong>in</strong>s a useful tool).<br />

• We have also considered an approach with explicit D.. degrees of freedom <strong>in</strong> the TPEP.<br />

This NNLO-D.. approach leads to results very similar to the ones at NNLO <strong>in</strong> the<br />

theory without isobars, with the exception of the partial waves that are sensitive to<br />

pionic scalar-isoscalar correlations like e.g. 3 D 3 • We conclude that the <strong>in</strong>clusion of the<br />

D.. via resonance saturation of dimension two 1f N LECs capture the essential physics of<br />

the isobar <strong>in</strong> the two-nucleon system. We note, however, that a more systematic study<br />

of pion-nucleon scatter<strong>in</strong>g <strong>in</strong> an EFT <strong>in</strong>clud<strong>in</strong>g the D.. is needed to further quantify<br />

these statements.<br />

4. F<strong>in</strong>ally, we have considered electromagnetic and strong isosp<strong>in</strong> violation <strong>in</strong> low-energy<br />

nucleon-nucleon scatter<strong>in</strong>g <strong>in</strong> the effective field theory formalism developed <strong>in</strong> ref. [91].<br />

We now summarize the results of this <strong>in</strong>vestigation.<br />

• We first considered isosp<strong>in</strong> violat<strong>in</strong>g parts of the effective Lagrangian. <strong>The</strong> terms<br />

related to the pion and the pion-nucleon system were discussed <strong>in</strong> [199], [222]-[224].<br />

lThis was also noted by Kaplan et al. [91].<br />

169

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!