The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

104 3. The derivation of nuclear forces from chiral Lagrangians
be described in detail below. For gA and i1f we will take the values known from the one-nucleon
sector. Note, however, that we have not performed here a complete renormalization ofthe potential
at NLO, which would also require a calculation of the contributions from disconnected diagrams
(zero- and one-nucleon operators). Therefore, the values of gA and i1f may, strictly speaking,
differ from the corresponding ones obtained from the 1fN scattering by order Q2 corrections and
should, in principle, be extracted from the N N scattering data. It would be interesting to perform
a complete renormalization of the effective potential to obtain more rigorously the values of these
constants. As we have found, the N N scattering data can be reproduced quite accurately using
the values i1f = 93 MeV and gA = 1.26.
Let us now make another comment concerning the choice of the loop functions JO,2,4, eq. (3.284).
Clearly, any other choice of defining this divergent loop integrals would give different finite subtractions
but leads to the same non-polynomial terms (3.300) in the potential. For example, to
express the potential in terms of
Ja = J2/L
T ' - 100 2/L 3
- r�o dl
J4 = 1 dl ,
(3.308)
we have to replace Ja, J2 and J4 by Ja + In 2, J2 + 2f-L2 and J4 + 4f-L4, respectively, in eqs.
(3.292)
(3.289),
and (3.297)-(3.299). This only modifies the definitions of the renormalized couplings
gA and Gi'S but does not change the expressions (3.300) and (3.270) for the effective potential.
Furthermore, to reproduce the results obtained using dimensional regularization we have to skip
the power-Iaw divergences, i.e. to set J2 = 0, J4 = O. Also in that case we get the same expressions
for the effective potential.
The effective potential at NNLO corresponding to eq. (3.255) can be renormalized analogously.
As can be seen from eq. (3.255), the contribution to the effective potential in the projection
formalism is the same as the one obtained with time-ordered perturbation theory (in the static
limit for nucleons). Furthermore, the related graphs shown in fig. 3.15 involve all possible time
orderings. As a consequence, the contribution to the effective potential can be obtained using
covariant perturbation theory and the technique of Feynman diagrams. This has been done by
Kaiser et al. using dimensional regularization [108]. Here, we will not perform explicit calculations
and simply adopt their result. The TPEP at NNLO reads:
where
(3) _
V21f,1-loop - VNNLO + P(k, q) , (3.309)
TPEP
-
VJJtJ -l��;i { -16m(���+ q2) + (2M;(2Cl -C3) -q2 (C3 + ::�)) (2M; + q2)A(q) }
128��ii (Tl ' T2) { -4�1��2 + (4M; + 2q2 -g�(4M; + 3q2))(2M; + q2)A(q)}
� �
+ 51;:�ii ((0\ . ij)(ih . ij) -q2(ih .52)) (2M; + q2)A(q)
3J:ii (Tl ' T2) ((51 ' ij)(52' ij) -l(5l .52))
X {(C4 + _1 4m _)(4M; + q2) - 8m
g� (10M; + 3q2)} A(q)
3g� . ( al � � ) ( � I �) (2M2 2)A()
641fmii z
+ a2 . P x P 1f + q q