The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

102 3. The derivation o{ nuclear {orces {rom chiral Lagrangians
with
(3.292)
Performing anti-symmetrization as described in appendix E allows to map the two spin-isospin operators
appearing in eq. (3.291) onto the two non-derivative operators used in v�o�, cf. eq. (3.269).
Therefore, as follows from eq. (E.8), the effect of the one-Ioop corrections to the lowest order contact
interactions with four nucleon legs can be completely absorbed by renormalization of the
constants C� and C� ,
Cs = C� -3S, Cf = C� -3S . (3.293)
We note that furt her renormalization of these couplings is due to the two-pion exchange, as
discussed below.
Consider now the proper two-pion exchange contribution (3.279). The first two integrals can be
immediately expressed in terms of the divergent loop functions JO,2,4 using h , h and h defined
in eqs. (D.2), (D.3) and (D.6). To calculate the last integral we use the following identity:
1 ä 1 (3.294)
Now it is easy to express the integrals entering the last two lines of eq. (3.279) in terms of the
loop functions JO,2,4. Specifically, we have:
(3.295)
Here, we have set q == Iql. Putting pie ces together, the expression for V2�:1-100P takes the form
V2�:1-100P
with
= vJl�p + (Sl + S2 q2) (7"1 . 7"2) + S3 [(0\ . q) (ih . if) - (51 . 52) q2] ,
1 { 2 4 2 M7r 2 4
2f
2
4 3847r 7r -18M7r(5gA -2gA) In - -M7r(61gA - 14gA + 4)
ft
+18M;(5g� -2g�)Jo -3(3g� -2g�)J2 } 1 { 4
,
2 M7r 1 4 2
+(23g� -lOg� - l)Jo },
3g� { M7r 1
647r2 -Jo }
f; ft 3 '
3847r2 f; (-23gA + 10gA + 1) In -;- - 2(13gA + 2gA)
- ln- + -
and the non-polynomial part vJlJP given by
(3.296)
(3.297)
(3.298)
(3.299)