The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

40 2. Low-momentum effective theories for two nuc1eons
V (VI) [GeV-2]
-2
-4
9
-6
-8
7
-10
5
-12
-14
3
[GeV]
q [GeV] q [GeV]
Figure 2.9: Left panel: effective two-nucleon potential VI (solid lines) in comparison with the
original potential V (dashed lines) for momenta less than 200 MeV. Right panel: difference between
the original and effective potential V - VI as a function of momenta q, ql .
Bere, V�ontact is a string of local contact terms of increasing dimension, which is caused by the
heavy mass particle and the high momenta p > A. We define the piece Vifght to be the light
meson exchange contribution (for ql, q < A): Vifght = Viight , w here Viight denotes the second term
in eq. (2.113). Specifically,
with
VI = V(O) + V(2) + V(4) + V(6) +
contact
V(O)
V(2)
V(4)
V(6)
Co ,
C 2 (q12 + q2)
C 4(q12 + q2)2 + C�q12q2 ,
. . . ,
C 6 (q12 + q2)3 + C� (qI2 + q2)q12q2 ,
(2.116)
(2.117)
and the superscript I (2n)' gives the chiral dimension (i.e. the number of derivatives). Thus the first
term on the right hand side of eq. (2.115) is the purely attractive part of the original potential due
to the light meson exchange. In this way, we have an effective theory for NN interactions, in which
the effects of the heavy meson exchange and the high momentum components are approximated
by the series of N N contact interactions and the light mesons are treated explicitly. Note that the
constants Ci , CI in eq. (2.117) correspond to renormalized quantities since the effective potential
VI (ql, q) is regularized with the sharp cut-off A. The first question we address in our model is:
what is the value of the scale Ascale and its relation to the cut-off A? Of course, a priori these two
scales are not related. For the kind of questions we will address in the following, we can, however,
derive some lose relation between these two scales as discussed below. Since we know VI (ql , q)