The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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