The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

5.5 Classification scheme 163
scattering length and effective range are exactly reproduced. The two new parameters E� I,2) are
determined from the nn and pp scattering lengths. The resulting parameters at f-l = Mn are of
natural size,
-3.46 fm2 , C2 = 2.75 fm4 , D 2 = 0.07 fm4 ,
-6.47 fm2 , E� 2) = 1.10 fm2 . (5.40)
To arrive at the curves shown in fig. 5.4, we have used the physical masses for the proton and the
neutron. We stress again that the effect of the em terms rv E� I, 2) is small because of the explicit
factor of a not shown in eq. (5.40). Having fixed these parameters, we can now predict the nn ISO
phase shift as depicted by the solid line in fig. 5.4. It agrees with nn phase shift extracted from
the Argonne V18 potential (with the scattering length and effective range exactly reproduced) up
to momenta of about 100 MeV. The analogous curve for the pp system (not shown in the figure) is
elose to the solid line since the CSB effects are very small. The leading CIB effect for the effective
range, ßrclB = 1/2(rnn + rpp) - rnp, is given by the last term in eq. (5.39). With the values of
the parameters shown in eq. (5.40) we obtain
ßrClB = 0.01 fm , (5.41)
which agrees with the small value observed experimentally for this quantity, see eq. (5.1).
5.5 Classification scheme
In the framework presented here, it is straight forward to work out the various leading (LO), nextto-leading
(NLO) and next-to-next-to-leading order (NNLO) contributions to CIB and CSB,
with respect to the expansion in Q, to leading order in a and the light quark mass difference. Here,
we will simply enumerate the pertinent contributions which appear at a given order. This also
naturally gives an estimate about their relative numerical importance based on simple dimensional
analysis. Let us first consider CIB. In the elassification scheme given below, TPE/3PE stands for
two / three-pion-exchange.
LO
aQ-2
NLO
a Q-l , c Q-l
NNLO
aQo
Pion mass difference in OPE,
four-nueleon contact interaction with no derivatives.
Pion mass difference in TPE and in the exchange of one radiation pion,
four-nueleon contact inter action with two derivatives,
insertions proportional to the np mass difference,
Pion mass difference in 3PE and TPE,
four-nueleon contact interaction with four derivatives,
CIB in the pion-nueleon coupling constants,
Note that there are no CIB effects due to the light quark mass difference linear in c = mu - md
from OPE and from the four-nucleon contact terms. These type of terms can only appear to
second order in c. Effects due to the charge dependence of the pion-nucleon coupling constants,
i.e. isospin breaking terms from .c��, only start to contribute at order aQo. Such effects are
therefore suppressed by two orders of Q compared to the leading terms, Le. by a numerical factor
of about (1/3)2. This finding is in agreement with the various numerical analyses performed in