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
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
2.3. Go<strong>in</strong>g to bigber energies: a toy model 43<br />
150<br />
100<br />
bO 50<br />
Q)<br />
�<br />
�<br />
0<br />
-50<br />
:<br />
'.<br />
��:' ,<br />
, , , , , ,<br />
"<br />
order 0 --- - --order<br />
2 -- - - -- - .<br />
order 4 --- --order<br />
6 _._._.exact<br />
--<br />
- -<br />
_._._.-::... .��<br />
bO<br />
Q)<br />
�<br />
�<br />
150<br />
100<br />
50<br />
0<br />
-50<br />
order 0 - - - ---order<br />
2 ,- ----..<br />
order 4 -----exact<br />
--<br />
-'-_-'- -- -- -1_-L_-' -- -- -'-_-'-_ L---<br />
-L.- --I<br />
-1 00 -1 00 L-o<br />
20 40 60 80 1 00 120 140 160 1 80 200 0 20 40 60 80 1 00 120 140 160 1 80 200<br />
T1ab [MeV] l1ab [MeV]<br />
Figure 2.11: Phase shifts from the effective potential V' (solid l<strong>in</strong>e) and the truncated expansion<br />
(2.116) as a function of the k<strong>in</strong>etic energy <strong>in</strong> the lab frame for A =300 MeV. Left (right) panel:<br />
the constants Ci, C; are fitted to the effecti ve potential (to the NN phase shift).<br />
with<br />
11 J 12 d 'O( ') TI (q', q,Eq)<br />
q q q, q ,<br />
E E +'<br />
q - q'<br />
ZE<br />
J 1 2 d I 1 2 d I T'* (q�, q,Eq)<br />
O<br />
( I ' ) T'(q�, q,Eq)<br />
h = q1 q1 q2 q2 E E<br />
. q1 ' q2 + '<br />
- ZE E q - E q� ZE<br />
(2.121)<br />
<strong>The</strong> results for quantum averages of the operators v(2n), (n = 0,1,2,3), are shown <strong>in</strong> table 2.3.<br />
q - q �<br />
Note that the matrix elements for the bound and scatter<strong>in</strong>g states have different units. This<br />
is a consequence of different normalization of those states. One observes a good convergence <strong>in</strong><br />
agreement with the naive expectation. Indeed, s<strong>in</strong>ce the cut-off is chosen to be A =300 MeV and<br />
the scale A s cale ,...,.,600 MeV, one expects the expansion parameter to be of the order AI A s cale � 1/2.<br />
Such a value agrees wen with the one extracted from the results shown <strong>in</strong> table 2.3. Note furt her<br />
that for higher energies, the convergence is slower, which is also rather natural.<br />
deuteron 25.94 MeV -4.23 MeV 1.07 MeV -0.38 MeV<br />
E1ab = 10 MeV 31.34 GeV-2 -6.19 GeV-� 1.64 GeV-� -0.58 GeV-2<br />
E1ab = 50 MeV 11.54 GeV 2 -3.28 GeV -",;! 1.11 GeV 2 -0.44 GeV 2<br />
E1ab = 100 MeV 7.79 GeV-2 -3.09 GeV-� 1.40 GeV-2 -0.71 GeV-2<br />
Table 2.3: <strong>The</strong> quantum averages of the operators V(O), V(2), V(4) and V(6) for the bound (second<br />
row) and the scatter<strong>in</strong>g states (third to fifth rows) for A = 300 MeV.<br />
Hav<strong>in</strong>g calculated the quantum averages of the operators V(O), V(2), V(4) and V(6) we can now<br />
better understand the quite slow convergence of the effective theory expansion (2.116) observed