The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.2. Two nucleons at very low energies
Here In and I(k) are defined by
Note that In and I(k) depend on A. Solving the linear matrix equation (2.11) one obtains
r (k) = � ( -Co - Ci(h + k 2 h t k 4 h � k 6 I(k)) C2( -1 + C2(h + k 2 h + k4 I(k))) )
w C2( -1 + C2(h + k h + k I(k))) -Ci(h + k 2 I(k))
with the constant w given by
w = - (-1
+ C2h) 2 + (h + k 2 I(k)) (Co + C2(C2h - k 2 (-2 + C2h)))
Using eq. (2.8) one finds for the on-shell T-matrix Ton(k) == T(k, k; k):
Ton (k) = fÄ (k 2 )
Co + C2(C2h + k 2 (2 - C2h))
(-1 + C2h)2 - (h + k2I(k))(Co + C2(C2h + P(2 - C2h)))
17
(2.13)
(2.14)
(2.15)
(2.16)
This is the final result for the on-shell T-matrix at next-to-Ieading order, which follows from our
effective theory. ll
We are now in the position to fix the unknown constants Co and C2 from the data. One way to do
that is to fit the phase shift b corresponding to eq. (2.16) to the experimental one. Alternatively,
one can require that the scattering length a and the effective range r, which are the leading two
coefficients in the effective range expansion for the S-wave phase shift
1 1
k cot(b) = --+ -rk 2 + v2k 4 + v 3 k 6 + v4k8 + O(k10) ,
a 2
(2.17)
are exactly reproduced. In this formula the v's denote the so-called shape parameters. We prefer
here the second way, since it allows for analytic expressions of Co and C2• For the derivation of the
effective range expansion of phase shifts in scattering theory the reader is referred to refs. [142],
[143]. We can easily obtain the effective range expansion for the T-matrix using the definition of
the on-shell S-matrix projected onto the S-wave:
Thus, the phase shift b can be expressed as
Applying the relation
with z = i - 2j(7rkmTon) to eq. (2.19) leads to
so n (k) = exp(2ib(k)) = 1 - i7rkmTon(k) .
b = ; i In (1 - i7rkmTon(k)) .
_ 1 (iZ - 1) In = arccot(z) ,
2i iz + 1
1 7rm
-T (kcotb-ik)
-- -- + - rk + V2k + V3k + V4k + O(k ) - zk
2 a 2
7rm ( 1 1 2 4 6 8 10 · )
11 At leading order one should keep in the potential (2.5) only the first constant term.
(2.18)
(2.19)
(2.20)
(2.21)