The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

5.1 Isospin violating effective Lagrangian 153
unnaturally large S-wave scattering lengths. This can be dealt with in different manners. One
is the approach proposed by Weinberg, in which chiral perturbation theory is applied to the
kernel of the Lippmann-Schwinger equation (effective potential). This approach is very similar
to the Hamiltonian formalism used in this work and explained in detail in chapter3. This route
was followed by van Kolck et al. , see refs [75], [74], [76], [78]. A different fashion to deal with
low-energy nucleon-nucleon scattering is the approach recently proposed by Kaplan, Savage and
Wise (KSW) [91].2 Essentially, one resums the lowest order local four-nucleon contact terms rv
Co (Nt N)2 (in the S-waves) to generate the large scattering lengths and treats the remaining effects
perturbatively, in particular also pion exchange. This means that most low-energy observables
are dominated by contact interactions. The chiral expansion for NN scattering entails a new
scale ANN of the order of 300 MeV, so that one can systematically treat external momenta up
to the size of the pion mass. There have been suggestions that the radius of convergence can
be somewhat enlarged [88], but in any case ANN is considerably sm aller than the typical scale of
about 1 Ge V appearing in the pion-nucleon sector . For recent calculations of various properties of
the two-nucleon system within this formalism see e.g. refs. [87], [88], [92], [95], [98], [210], [211]. In
this context, it appears to be particularly interesting to study CIB (or in general isospin violation)
which is believed to be dominated by long range pion effects. That is done here.3 First, we write
down the leading strong and electromagnetic four-nucleon contact terms. It is important to note
that in contrast to the pion or pion-nucleon sector, one can not easily lump the expansion in
small momenta and the electromagnetic coupling into one expansion but rat her has to treat them
separately.4 Then we consider in detail CIB. The leading effect starts out at order aQ-2, where Q
is the generic expansion parameter in the KSW approach. It sterns from OPE plus a contact term
of order a with a coefficient aE� l ) of natural size that scales as Q-2. Similarly, the leading CSB
effect results from contact terms with four nucleon legs of order a and order E == mu - md, which
also scale as Q-2. While in the case of E�l) this scaling property is enforced by a cancellation
of a divergence, the situation is a priori different for CSB. However, for a consistent counting of
all isospin breaking effects related to strong or electromagnetic (ern) insertions, one should count
the quark mass difference and virtual photon effects similarly. Note, however, that these CIB and
eSB terms are numerically much smaller than the leading strong contributions which scale as
Q-l because a« 1 and (mu - md)/Ax «1. The corresponding constants, which we call E�1,2),
together with the strong parameters (as given in the work of KSW) can be determined by fitting
the three scattering lengths a pp , ann, an p and the np effective range.5 That allows to predict the
moment um dependence of the np and the nn 1 So phase shifts. Based on these observations, we
can in addition give a general classification for the relevant operators contributing to eIB and
eSB in this scheme. Additional work related to long-range Coulomb photon exchange is necessary
in the proton-proton system. We do not deal with this issue here but refer to recent work using
EFT approaches in refs. [219] , [220], [221].
5.1 Isospin violating effective Lagrangian
Let us first discuss the various parts of the effective Lagrangian underlying the analysis of isospin
violation in the two-nucleon system. To include virtual photons in the pion and the pion-nucleon
2 There exist by now modifications of this approach and it has been argued that it is equivalent to cut-off schemes.
We do not want to enter this discussion here but rather stick to its original version.
3For a first look at these effects in an EFT framework (based on the Weinberg power counting) , see the work of
van Kolck [75]. Electromagnetic corrections to the one-pion exchange potential have been considered in [218].
4 This is because of the different power counting schemes in these cases.
5Whenever we talk of the pp system, we assume that the Coulomb effects have been subtracted.