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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- i�03{ [(NtVN) . (VNt x ifN) + (VNtN) . (Ntif x VN)]<br />
- (NtN)(VNt . if x VN) + (NtifN) · (VNt x VN)}<br />
1 1 - -<br />
- 2 ( "2 C4 (8ik8jl + 8il8kj) + C58ij8kl )<br />
x [(oiojNtakN) + (NtakOiOjN)] (NtaIN)<br />
- (�06 (8ik8jl + 8il8kj) + (05 - 67)8ij8kl) (NtakOiN)(ojNtaIN)<br />
1 ( 1 - - - )<br />
- 2 2 (C4 - C6) (8ik8jl + 8il8kj) + C78ij8kl<br />
x [(OiNtakOjN) + (OjNtakoiN)] (NtaIN) .<br />
189<br />
(F.21)<br />
<strong>The</strong> effective Lagrangian (F.21) is sufficient for the derivation of the short-range part of the twonucleon<br />
potential at NLO. To obta<strong>in</strong> the NNLO potential one needs the Lagrangian .c�� for<br />
contact <strong>in</strong>teractions with four nucleon legs. Let us first discuss the terms with three derivatives<br />
act<strong>in</strong>g on the nucleon field Hv (Hv) .<br />
• Terms with 0/-t, ov, 0p.<br />
It is, c1early, not possible to construct a Lorentz scalar of the form<br />
(F.22)<br />
if the operators f 1 ,2 conta<strong>in</strong> only three derivatives and no velo city or sp<strong>in</strong> operators (or, <strong>in</strong><br />
general, if an odd number of the operators 0/-t' Vv and S(J enters f1 and f2) .<br />
• Terms with 0/-t, ov, op and S(J.<br />
<strong>The</strong> terms without <strong>in</strong>sert ions of the totally anti-symmetrie tensor f et ß7 (J are not parity<br />
<strong>in</strong>variant. For the terms with f et ß7(J we first note that the derivatives must always act onto<br />
different fields Hv (Hv) beeause of the anti-symmetrie properties of the Levy-Civita tensor.<br />
<strong>The</strong> only two terms one ean build up are:<br />
f /-t VP (J { (Hv *a /-t 8 vHv )(Hv *a pS(JHv) + (Hv *a v 8 /-tHv)(Hv 8 pS(JHv)} , (F.23)<br />
if /-t VP (J { (Hv *a /-t 8 vHv)(Hv *a pS(JHv) - (Hv *a v 8 /-tHv)(Hv 8 pS(JHv)} . (F.24)<br />
<strong>The</strong> seeond term vanishes after apply<strong>in</strong>g partial <strong>in</strong>tegration. We will now show that the<br />
first term ean be elim<strong>in</strong>ated from the Lagrangian us<strong>in</strong>g the EOM for the nucleon field and<br />
eqs. (F.2), (F.16) . Let us eonsider only the first term <strong>in</strong> eq. (F.23) to keep our notation<br />
more eompaet:<br />
f /-t Vp(J (Hv *a /-t 8 vHv)(Hv *a p S(JHv)<br />
= f /-t VP (J ( Hv *a /-t 8 vHv ) ( Hv *aet [vp Vet S(J - 2(Sp Set S(J + Set Sp S(J)] Hv)<br />
(F.25)<br />
= -2 f /-tv<br />
P (J ( Hv *a /-t 8 vHv) ( Hv *aet (i fpetß7 vß S7 S(J + Set {Sp, S(J}) Hv) + . ..<br />
. - � ---ct ( ) ( - �et = 2 z det(R) Hv a /-t (j vHv Hv 0 v ß S 7 S(JHv ) + ... ,