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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122 4. <strong>The</strong> two-nuc1eon system: numerical results<br />
4.2 <strong>The</strong> fits<br />
In this section we, basicaIly, concentrate on the determ<strong>in</strong>ation of the various coupl<strong>in</strong>g constants.<br />
For Mn:, <strong>in</strong>: and gA we use the follow<strong>in</strong>g values:<br />
Mn: = 138.03 Me V , <strong>in</strong>: = 93 MeV , gA = 1.26 . ( 4.14)<br />
Furthermore, as already stated above, the values of C 1 ,3,4 are<br />
Cl = -0.81 GeV -1 , C3 = -4.70 GeV-1 , C4 = 3.40 GeV-1 . ( 4.15)<br />
Thus, all parameters enter<strong>in</strong>g the non-polynomial part of the potential are fixed.<br />
Consider now the polynomial part vcontact of the potential represented by contact <strong>in</strong>teractions<br />
vcontact = V(O) + V(2). To p<strong>in</strong> down the n<strong>in</strong>e parameters Cs, CT, Cl , ... ,C7 we do not perform<br />
global fits as done <strong>in</strong> ref. [78]. Rather we <strong>in</strong>troduce the <strong>in</strong>dependent new parameters C2s+1l.,<br />
J<br />
02S+1l. via the follow<strong>in</strong>g equations:<br />
J<br />
v contact (3 SI )<br />
41f (Cs - 3CT) + 1f (4C1 + C2 - 12C3 - 3C 4 - 4C6 - C7)(p2 + p'2)<br />
- 2 2<br />
C1S0 + C1S0 (p + p' ) , (4.16)<br />
41f (Cs + CT) + i (12C1 + 3C 2 + 12C3 + 3C4 + 4C6 + C7)(p2 + p'2)<br />
- 2 2<br />
C3S1 +C3S 1 (p + p' ), (4.17)<br />
2 ; (-4C1 + C2 + 12C3 - 3C4 + 4C6 - C7) (p p') = Cl PI (pp') , (4.18)<br />
2 ; (-4C1 + C2 - 4C3 + C4 + 2C5 + 4C6 + C7) (p p')<br />
C3P1 (pp') ,<br />
2 ; (-4C1 + C2 - 4C3 + C 4 + 2C5) (pp') = C3P2 (pp') ,<br />
21f<br />
3 (-4C1 -<br />
+ C2 4C3<br />
) ,<br />
+ C4 + 4C5 + 12C6 - 3C7 (pp )<br />
C1 Po (pp') ,<br />
2.j21f ( ,2 ,2<br />
- 3 - 4C6 + C7) p = C3Dl-3S1 P ,<br />
2.j21f 2 2<br />
- 3 - (4C6 + C7)p = C3Dl-3S1 P .<br />
(4.19)<br />
( 4.20)<br />
(4.21)<br />
(4.22)<br />
(4.23)<br />
Here, Ves+1lj) denotes the matrix element (lsjlVllsj) and Ves+1lj _2s+1lj) is (lsjlVll'sj). <strong>The</strong><br />
equations (4.16)-(4.23) can be obta<strong>in</strong>ed with the help ofthe formulae of appendix G for the partial<br />
wave decomposition. <strong>The</strong> ma<strong>in</strong> advantage of us<strong>in</strong>g the new parameters C2s+1[., 02s+1l. <strong>in</strong>stead of<br />
J J<br />
the old ones Cs, CT and Cl , ... , C7 is that now we can fit each partial wave separately. This not<br />
only makes the fitt<strong>in</strong>g procedure extremely simple, but also leads to unique results. To lead<strong>in</strong>g<br />
order, the two S-waves are depend<strong>in</strong>g on one parameter each. At NLO, we have one additional<br />
parameter for 1 So and 3 SI as weIl as one parameter <strong>in</strong> each of the four P-waves and <strong>in</strong> EI' At<br />
NNLO (NNLO-Ll), l we have no new parameters, but must refit the various contact <strong>in</strong>teractions<br />
due to the TPEP contribution <strong>in</strong> all partial waves. We have used two different methods to fix the<br />
1We will denote by NNLO-ß the potential<br />
(4.24)