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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120 4. <strong>The</strong> two-nuc1eon system: numerical results<br />
where<br />
and q=p' - p.<br />
4 39� r i (0\ + eh) . (p ' x ff) (2M; + q2)A(q)<br />
6 7fm 'Ir<br />
647fm f: Tl' T2 2 0"1 + 0"2 •<br />
x {(c4 + 4 �)(4M; + q2) - :� (10M; + 3q2)} A(q)<br />
g � (1 - g �) ( ) . ( � � ) ( �' ;;'\ (4M2 2)A( )<br />
L(q)<br />
A(q)<br />
q V 'Ir + q n<br />
1 q<br />
P X p) 'Ir + q q ,<br />
� . /4M2 2 I V 4M; + q2 + q<br />
2M'Ir '<br />
-arctan --<br />
2q 2M'Ir '<br />
Now we want to apply this potential to calculate the two-nucleon observables. <strong>The</strong> scatter<strong>in</strong>g<br />
states are described by the Lippmann-Schw<strong>in</strong>ger <strong>in</strong>tegral equation. <strong>The</strong> LS equation (for the<br />
T-matrix) projected onto states with orbital angular momentum I, total sp<strong>in</strong> s and total angular<br />
moment um j is<br />
InOO d " ,,2<br />
sj , sj , P P sj ,,, m sj<br />
Tl'<br />
"<br />
I(P , p) = Vi, I (P , p) + L<br />
(2 )3 Vi, 1" (P , p ) 2 ,,2 . Tl" , ,<br />
I (P , p) ,<br />
I" ° 7f ' P - P + 2'TJ '<br />
with 'TJ --+ 0+. In the uncoupled case, I is conserved. <strong>The</strong> partial wave projected potential Vi��(p' , p)<br />
can be obta<strong>in</strong>ed us<strong>in</strong>g the formulae of the appendix G. <strong>The</strong> relation between the on-shell 8- and<br />
T-matrices is given by<br />
S � ( )_ r<br />
2<br />
T� ( )<br />
1'1 P - Ul'l - 87f2 pm 1'1 P ,<br />
where P denotes the two-nucleon center-of-mass three-momentum. <strong>The</strong> phase shifts <strong>in</strong> the uncoupled<br />
cases can be obta<strong>in</strong>ed from the S-matrix via<br />
SOj ( 2 ' rOj )<br />
JJ J ' Slj (2·r1j)<br />
.. = exp 2u ·<br />
(4.4)<br />
(4.5)<br />
(4.6)<br />
jj = exp w j , (4.7)<br />
where we have used the notation 8t j . Throughout, we use the so-called Stapp parametrization<br />
[203] of the S-matrix <strong>in</strong> the coupled channels (j > 0):<br />
S =<br />
)<br />
2 . sm<br />
.<br />
(2 f ) exp ( W' r1j<br />
j_ 1 ;-. 2Uj+1<br />
. r1j<br />
) .<br />
(4.8)<br />
cos (2f) exp (2i8 j� 1)<br />
For the discussion of the effective range expansion for the 3 S I partial wave we will use the different<br />
parametrization of the S-matrix, namely the one due to Blatt and Biedenharn [204]. <strong>The</strong><br />
connection between these two sets of parameter is given by the follow<strong>in</strong>g equations:<br />
8j-1 + 8j+1<br />
s<strong>in</strong>(8j_ 1 - 8j+1 )<br />
8j-1 + 8j+1 ,<br />
tan(2f)<br />
tan(2E') ,<br />
s<strong>in</strong>(2f)<br />
s<strong>in</strong>(2E') ,<br />
(4.9)