The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.2. Two nuc1eons at very low energies 21
In quantum field theories dimensional regularization (DR) appears to be a simple and powerful
calculational tool, which allows to preserve all known symmetries. We will now try to apply it to
the low-energy 1. effective theory for two nueleons. Since we will not use the cut-off we should put
JA(p2) = Further, all integrals In have to be generalized to an arbitrary number of dimensions
d:
1000 d3q 1000 ddq
I - -2m7f2 -- q n __
-3 ---+ -2m7f2J-L3-d q n-3
n -
(2.49)
0 (27f)3 0 (27f)d '
where f.L is the mass scale introduced due to the regularization. It turns out that all power law
divergences like those in eq. (2.49) vanish after performing the dimensional regularization, see, for
example, [82]. This will be of crucial importance for our furt her considerations. Thus, we obtain
a very simple expression for the inverse
1
of the T-matrix from eq. (2.16):
(2.50)
Again, we can fix the constants Co and C2 requiring that the first two terms in the effective range
expansion (2.21) of the on-shell T-matrix are exact reproduced. This lead to
Co = 2a
7fm
,
(2.51)
The inverse T-matrix (2.50) with the constants Co, C2 given by eq. (2.51) maps to the effective
range expansion only for:
(2.52)
This has to be compared with the corresponding result (2.47) for cut-off regularization. Dimensional
regularization provides a very small domain of validity of the effective theory if the
scattering length is large. For the case a ---+ 00 such a theory even becomes completely useless.
The failure of dimensional regularization can easily be understood if one considers a simple model
[83] with an S-wave separable potential given by
2) -1/2
( 2 12) -1/2 ( V (pI, p) = - : 1 + � 1 + 7f m 2 � 2 ' (2.53)
where 9 is a dimensionless coupling and A is a scale that characterizes the range of the interaction.
The on-shell T-matrix can be obtained from the LS equation (2.6),
Ton(k) = V(k, k) [1 - m (oo q2 dq k2 V(q ; q) . ] -1 = _� [�(1 + A k :) - (A + ik)] -l . (2.54)
io - q + ZE 7fm 9
Comparing this result with the effective range expansion (2.21) allows to read off the scattering
1 length and effective range:
-1 9 2
- Ag ' r = gA . (2.55)
Both quantities are inversely proportional to the range of the potential A, as it follows from
dimensional reasons. Further, the scattering length takes an unnatural large value if the coupling
constant 9 is elose to one. This can be seen as a result of the cancelation between the two
terms in the first bracket in eq. (2.54). The first term results from the tree-contribution to the
amplitude, whereas the second one comes from virtual excitations in the loops, see fig. (2.2). In the
a - - -