The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

186 F. The complete set of the N N contact interactions with two derivatives
There are also two terms with one derivative:
Note that the terms with just one insertion of the spin-operators S/1 like, for instance,
are not parity and charge conjugation invariant, see table F.l. The same holds true for the term
(F.8)
(F.9)
(F.10)
Both terms in eq. (F.8) are redundant and can be eliminated from the effective Lagrangian applying
the leading order equation of motion (EOM) (3.165) of the field Hv : (v · ö)H = 0. 6 Note, however,
that the terms in (F.8) are reparametrization invariant. This can easily be checked using eqs. (F.4)
and (F.6).
Let us now consider the contact interactions with two derivatives. The following terms appear in
the effective Lagrangian:
.c�� = �Ü1 [ (Hv 7J /1 Hv)(Hv 7J/1 Hv) + (Hv 8 /1Hv)(Hv 8/1 Hv )]
+ ü 2 (Hv 7J /1 Hv )(Hv 8/1 Hv )
+ ü3(HvHv)(Hv( 82 + 7J2 )Hv)
+ ü4(HvHv)(Hv 8 /1 7J/1 Hv)
1 [ - ---itv - ,="p - ,="v - ---itp ]
+ 2i Ü5 E/1VpuV /1 (Hv iJ Hv )(Hv 'ä SU Hv ) - (Hv 'ä Hv)(Hv SU iJ Hv)
+ i Ü6 E/1vpuv /1 (HvHv)(Hv 8v s p7Ju Hv)
- - '="P---itu
+ i Ü7 E/1vpuv/1(HvSV Hv)(Hv 'ä iJ Hv)
+ �i ÜS E/1VpuV /1 [(Hv 7Jv Hv )(HvS p7Ju Hv) - (Hv 8v Hv )(Hv 8u S P Hv)]
1
+ 2 (ügg/1Pgvu + ü10g/1ugvP + üllg/1vgpu)
X [(HvS pa/1 Hv)(Hvs u7Jv Hv) + (Hv 8/1 S P Hv)(Hv 8V sU Hv)]
+(Ü1 2 g/1pgvu + Ü13g/1ugvp + Ü14g/1vgpu ) (HvS p7J/1 Hv )(Hv 8v s u Hv )
+ � (�Ü15 (9/1Pgvu + g/1ugvp) + Ü169/1V9pu)
x [(Hv 8/1
s
p7JV
Hv)(HvSU Hv) + (Hv 8v
sP7J/1 Hv )(HvSU Hv)]
1 1 ( ) - '="/1'="V ---it/1---itv P ---itv -
(F.ll)
+ 2 2Ü17(9/1Pgvu + g/1ugvp) + Ü1Sg/1vgpu (Hv( 'ä 'ä + iJ iJ )S iJ Hv)(HvS Hv)
where Ü1, ... ,lS are constant coefficients. Here we have not shown terms which include the operator
v . Ö, since they can be eliminated applying the equation of motion (3.165) for the field Hv .
6The EOM (3.165) is only valid modulo higher order terms. If no pion fields are considered, such higher order
terms may be divided into two classes: the on es with just one field operator Hv and those ones involving more field
operators (like, for instance, the term (HvHv)Hv). The leading correction of the first kind is given by the term
-iß2/(2m). The higher order corrections to eq. (3.165) of the second kind would lead to contact interactions with six
and more nucleon legs and are irrelevant for our discussion. Thus, eliminating the terms (F.8) using the EOM for Hv
would modify the contact interactions with four nucleon legs at order .6.i = 3 (terms like l/m(Hv 7J. a Hv )(HvHv)).
For further discussion on such redundant terms see e.g. refs. [133], [71].
U