The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

Appendix H
Formulae for the deuteron properties
Here, we collect the formulae needed to calculate the deuteron properties. We denote by u( r)
and w(r) the S- and D-wave coordinate space wave functions, further, the momentum space
representations of u(r)/r and w(r)/r by ü(p) and w(p), respectively. We have
N ormalization
D - state probability
Quadrupole moment
Asymptotic S - state
Asymptotic D /S - ratio T}
RMS (matter) radius :
1000 dpp2 [ü(p)2 + w(p)2] = 1000
1000 dpp2 w(p)2 = 1000 dr w(r)2 ,
Qd = � (oo drr2 w(r) [VSu(r) -w(r)]
20 Jo
dr [u(r)2 + w(r)2] = 1 , (H.1)
= _� (oo dP{VS[p2dÜ(P) dw(p) + 3pw(p) dü(P)]
20 Jo dp dp dp
(H.2)
+p2 (d��)) 2 + 6w(p)2 } , (H.3)
u(r) ----7 As e - ,,( T for r ----7 00 , (H.4)
w(r) ----7 T} As (1 + ;r + h : )2) e- ,,(T for r ----7 00 , (H.5)
rd = l [1000 drr2 [u(r)2 + w(r)2] f/2 , (H.6)
with r = Jm IEdl = 45.7MeV (using m = (mp + mn)/2). Note that the momentum-space representation
of Qd given in eq. (H.3) shows why one cannot use a sharp momentum-space regulator
to calculate this quantity. We also remark that the D-state prob ability is not an observable.
Meson-exchange current corrections to Qd are not given.
194