The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

44
2. Low-momentum effective theories for two nucleons
for the two-body binding energy as indicated in table 2.2. Indeed, the binding energy is given by
E(i) = - (w(i) IHo lw(i)) - (w(i) IViigh tlw(i)) (2.122)
- (w(i) IV(O) lw(i)) - (w(i) IV(2)lw(i) ) - ... - (w(i)lV(i)lw(i) ),
where Ho is the free Hamiltonian and the superscript i denotes the order of calculation. Now
putting the correct wave function (qlw) == (qlw(oo)) obtained from the unitarily transformed
potential instead of (qlw(i)) we end up using the values of (WIV(i)lw) from table 2.3 with the
following numerical estimation:
E = -9.3
MeV + 33.9 MeV
-25.9 MeV + 4.2 MeV - 1.1 MeV + 0.4 MeV + ...
2.3 MeV .
(2.123)
From this estimations we see that the rat her small value for the binding energy results from the
cancelation of various terms in the expansion (2.122). This is a similar sort of cancelation to that
observed for the scattering length of the model potential eq. (2.53). In fact, the convergence for
the binding energy is not slow but simply moved to higher order: whereas already the leading
order (LO) term (w(O)IV(O) lw(O)) provides a good approximation (±1O%) for the matrix element
(wIVcontactlw) , only the NNLO result for the binding energy shows a similar quality. Thus, we
expect a fast convergence, like those for the terms (w(i) lV(i) Iw(i)), for the binding energy starting
from NNLO. For instance,
IE - E(6)1
'" 0.21 .
(2.124)
IE -E (4) I
In the real world to which one applies the effective field theory, one does not know the true effective
potential V' and therefore also the constants Ci , CI are unknown. They have to be determined by
fitting such a type of effective potential to the N N data, like the low energy phase shifts andj or the
deuteron binding energy. We can perform this exercise also in our case. Keeping again Viight (q' , q)
as a separate piece we have determined by trial and error the constants Co, C 2 , •
. • by
solving the
inhomogeneous LS equation and truncating the series at various orders. We have performed three
different fits with V(O), V(O) + V(2) and V(O) + V(2) + V(4) representing the LO, NLO and NNLO
results, respectively. To achieve stable values for C i 's we have performed the fits below 10 Me V at
LO and below 25 and 100 MeV at NLO and NNLO.20 The results for the values of the coupling
constants Co, C 2 , C 4 and C� are summarized in table 2.4. We have not performed the sixth
LO 9.49 (9.63) - - -
NLO 10.91 (11.01) -23.09 (-24.80) - -
NNLO 10.82 (10.86) -25.79 (-27.74) 104.1 (121.9) -251.7 (-253.9)
Table 2.4: The values of the constants C i determined from fitting the phase shift (to the effective
range parameters).
order (N3LO) fit since already the fourth (NNLO) order result gives an excellent reproduction
of the phase shifts as shown in the right panel of fig. 2.11. Alternatively, one can determine the
C i 's requiring that the first terms in the effective range expansion (2.17) of the phase shift are
2° This is different to the procedure of the ref. [124], where a larger range of energies was used to fix the coupling
constants. Therefore the values of C;'s given here and in [124] are slightly different.