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