The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.8. Two-nucleon potential
Here,
g� (1 - g�) ( ) .
641fmf� Tl ' T2 Z (Tl + (T2 . P x P 1f + q q .
( � � ) ( � I �) (4M 2 2)A( )
A(q) =
1 q
2q 2M1f
- arctan -- .
Furt her , P(k, if) in eq. (3.309) is a polynomial in momenta of at most second degree and has the
same structure as the expressions in eq. (3.270).44 Thus, its explicit form is of no relevance here,
since it only contributes to the renormalization of the couplings Cs , CT , Ci . Again, no explicit
dependence on the renormalization scale appears in the expression (3.310) für the non-polynomial
terms. In ref. [108] it is also pointed out that all NNLO one-pion exchange diagrams containing
vertex corrections do not yield any pion-nucleon form factor and only lead to renormalization of
the 1fN coupling gA (i.e. they have the same form as the proper one-pion exchange). Clearly, the
same holds true for all NNLO graphs representing vertex corrections to short range interactions.
For the LECs Cl,3,4 entering eq. (3.310) we should take the values obtained from fitting 1fN phases
in the threshold region, see e.g. ref. [71] or, alternatively, from fitting the invariant amplitudes
inside the Mandelstarn triangle, i.e. in the unphysical region [195]. The so determined parameters
are only slightly different, but these small differences will play a role later on. For example, the
LECs CI,3,4 from fit 1 of ref. [71] are Cl = -1.23 GeV-l, C3 = -5.94 GeV-l and C4 = 3.47 GeV-l.
A recent investigation of the subthreshold amplitudes [195] leads to slightly different values, Cl =
-0.81 GeV-l, C3 = -4.70 GeV-l and C4 = 3.40 GeV-l. It is this latter set we will use in the
following. These values are also consistent with the re cent determination from the proton-proton
interaction based on the chiral two-pion exchange potential [198].
Kaiser et al. also included l/m corrections into the expression (3.310) for the two-pion exchange
potential an NNLO. This is consistent with the power counting scheme in the one-nucleon sector.
In our power counting scheme with Q/m rv Q2 / A�, where Q corresponds to the low moment um
scale, such l/m terms appear first one order higher. Nevertheless, we have decided to keep these
corrections, since otherwise one cannot directly use the values of the LECs Cl,3,4 as determined
from the 1fN sector in the presence of the l/m terms. By doing so we only include some sm aller
terms of higher order in the potential. It can easily be checked, that those terms are indeed small
as compared to the Cl,3,ccontributions. For example, the constant C4 enters eq. (3.310) only in
the combination C4 + 1/ ( 4m). Correspondingly, the pertinent C4 contribution is numerically more
than 10 times larger than the l/m correction.
To end this section we would like to point out once more that eqs. (3.269), (3.300), (3.270) and
(3.310) define the unique expressions for the effective potential up to NNLO, i.e. the results do
not depend on the regularization scheme, as discussed above.
105
(3.310)
(3.311)
3.8.3 Phenomenological interpretation of some of the 1f N LEes and the role
of the ß(1232)
It is well known that the values of some of the LECs can be understood phenomenologically in
terms of the resonance saturation hypotheses. This is discussed in detail in the reference [199]
for the purely meson sector and in ref. [200] for pion-nucleon interactions. The idea of such
a phenomenological interpretation is very simple. Consider the most general chiral invariant
Lagrangian for pions, nucleons and their excitations. The resonances can be integrated out from
44 More precisely, after performing the partial wave decomposition, fCk, if) leads in each partial wave to polynomials
in p and pi of at most second degree.