The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

18 2. Low-momentum effective theories for two nucleons
Let us now consider the on-shell T-matrix (2.16). The imaginary part of its inverse is given by
Here we have used the equality
k2 _
P i7r
q2 - 2k
i
= -7rkm .
2
8(k - q) ,
(2.22)
(2.23)
where P stays for the principal value integral. Thus, the imaginary parts of the T-matrix in
eqs. (2.21) and (2.16) agree automatically. This has to be expected from the unitarity of the Smatrix.
Before fixing the constants Co and C 2 we have to define the low-momentum expansions
of iX(k2) and of the principal value integral P(I):
1 + hk2 + i 4 k 4 + O(k6) ,
m (to + t 2 k2 + t4 k4 + O(k6))
(2.24)
(2.25)
where fi and ti are some constants. No negative powers of k2 in eq. (2.25) are allowed since
otherwise P (fo oo dq fÄ ( q2) / (k2 - q2)) would not exist for k = O. Expanding the inverse of the
T-matrix (2.16) and comparing the first two terms of this expansion with those in eq. (2.21) we
obtain the following two conditions for Co and C 2 :
7rm
2a
7rmr
4
Solving the second equation leads to two solutions for C 2 :
where
C 2 = � ±
2ah + 7rm
13 hVi5
D = (7r2m2 + 2am7r(2h - hh) + a2( 41l + rm7r h - 4mhto))
For the constant Co one obtains from eq. (2.26)
C _
Is (2ah + m7r)( -2a1j + 2ahIs + Ism7r ± 2Vi5Is)
_ _
0 - 12 3 D12 3
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
Let us now consider the conditions, under which real solutions for the constants Co and C 2 exist.
For that we should require
D > 0. (2.31)
From the definition (2.13) we see that
(2.32)
where 0: is non-negative dimensionless constant. This simply follows from dimensional reasons.
For the expansion coefficients tn, in in eqs. (2.24), (2.25) we have:
1
in