The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.2. Two nucleons at very low energies 19
For large A the term a2rm7fh in the expression (2.29) becomes dominant. Obviously, for positive
effective range r > 0 an upper bound for the values of A exists, above which no real solutions
for Co and C2 are available. This statement is quite general but not new: it can be derived from
Wigner's theorem [84], which is based on such general principles like causality and unitarity. The
theorem says that the rate dJ(k)jdk by which the phase shift can change with energy is bounded
from below by some function of the range R of the potential, the moment um k and the phase
shift J. Thus, since the effective range in the 1 So and 3 SI channels is positive, one cannot take
the cut-off A to infinity. At first sight, this might look like a failure of the cut-off regularization.
Indeed, the regularization scale (cut-off) is usually taken to infinity if the standard renormalization
technique is used for renormalizable field theories. However, in an effective theory the cut-off
becomes physically relevant. Introducing a cut-off in an effective theory provides finiteness of
the amplitude. At the same time, it leads to suppression of all high-moment um virtual states,
for which no information is available. That is why in an effective theory one should take the
cut-off below the scale, at which new physics is expected to appear. In the low-energy nucleonnucleon
effective theory such new physics is associated with pions. We cannot expect that our
derivative expansion of the potential will correctly represent effects of pion exchanges at energies
rv Mn . Thus, if we would take the cut-off A much larger than the pion mass Mn , we would simply
incorporate wrang physics. This explains why the cut-off should remain finite and of the order of
Mn ·
Up to now we have only considered a general scattering problem in the S-channel. We have
neither restricted ourselves to a concrete regularization scheme nor to a particular partial wave.
Let us now consider the 1 So channel and compare different regularization schemes for that case.
We will consider cut-off and dimensional regularizations. Let us start with the cut-off theory and
specify the function JA (p2). For simplicity, we will take
This leads to
(k) = -i7fm m 1 A + k
I
2k + 2k n A - k .
Further, one finds for the particular choice (2.34) of the function JA(p2) that
Ii = 0, ti = (i + l )Ai+l 1
.
For the constants Co and C2 one obtains
with
Co
9D + m(7f - 2aA) (m(97f - 8aA) ± 18VD)
5DAm
- 3 ( m (7f - 2aA))
mA3 1 ± VD '
D = �m2 (37f2 - aA (127f + aA(7frA - 16))) .
For the inverse of the T-matrix we get from eq. (2.16) a quite simple expression:
1 = Am (1 + (7f - 2aA)2 ) _
To n (k) 2aA(7f - 2aA) - a2(4 - 7frA)k2
I(k)k2 .
(2.34)
(2.35)
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)