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