The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

182 E. Anti-symmetrization of the contact interactions
Now it is easy to perform anti-symmetrization of eq. (E.1):
where
1
4 (a1 - a2 - a3 - 3(4) ,
1
4( -al - 3a2 + 5a3 + 3(4) . (E.6)
Note that only two independent constants ß1 and ß2 enter the expression (E.5) for the antisymmetrized
contribution. Therefore, we can choose only two operators of the four contact terms
in eq. (E.1) as the basis. For example,
Vo = C s + Cr eh . eh . (E.7)
Requiring that the anti-symmetrized contributions of eqs. (E.1) and (E.7) are the same we obtain
using eqs. (E.5), (E.6)
Cr (E.8)
One can proceed in the same way to consider the contact interactions with two derivatives. Altogether
fourteen such interactions contribute in the two-nucleon cms system (seven terms in
eq. (3.270) plus the the corresponding ones with insert ions of the isospin matrices). For our purpose
it is sufficient to consider a subclass of these interactions, namely those depending on q 2, k2
(since the other contact terms requiring anti-symmetrization enter eq. (3.296)):
V2 =
,1
q 2 + ,2 ih . eh q 2 + ,3 Tl . T2 q 2 + ,4 (0\ . eh) (T 1 . T2) q 2
+ ,1 k 2 + ,2 ih . ih k 2 + '3 Tl ' T2 k 2 + '4 (51 . (2) (Tl ' T2) k 2 .
Note that A[g] = -2k, A[k] = -g/2. Therefore, we obtain using eq. (E.4):
v2 W1 q - 12 W1 + W3 + W4 0"1 ' 0"2 q + W2 Tl . T2 q .
(E.9)
A 2 1 ( 4 3 ) - - 2 2 (E 10)
1
-
12
( 4W2 + W3 - W4) (51 . (2) (Tl ' T2) q2 + W3 k2 -1 ( W3 + 4W1 + 12w2 ) 51 . 52 k2 ( W4 + 4W1 - 4W2 ) (51 . 52 ) (Tl . T2) k2 ,
+ W4 Tl . T2 k2 -1
where W1,2,3,4 are given by
1 1
2'1 - 32 (,5 + 3,6 + 3'7 + 9,8 ) ,
1 1
2'3 - 32 (,5 + 3,6 - ,7 - 3'8 ) ,
1
2(,5 - ,1 - 3'2 - 3'3 - 9'4 ) ,
1
2(,7 - ,1 - 3'2 + '3 + 3'4) .
(E.ll)