The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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