The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

Chapter 2
Low-momentum effective theories for
two nucleons
2.1 What is effective?
All known interactions in nature can be classified in terms of four fundamental types: strong,
weak, electromagnetic and gravitational. The first three of them have been unified and build the
basis of the Standard Model. The crucial role in the development of this model has been played
by the requirement of renormalizability. One can easily check the property of renormalizability of
a theory by introducing a parameter ßi characterizing an interaction of type i, which contains fi
(bi) fermionic (bosonic) fields. In four dimensions it is defined by
ß· = 4-d·--f· -b·
2 - 2
3
2 2 2 , (2.1)
where di is the number of derivatives. This quantity simply defines the mass dimension of a
related coupling constant.1 This allows for the following classification of all interactions: those
with ßi 2: 0 are called renormalizable, whereas interactions with ßi < 0 are non-renormalizable
[132]. One defines furthermore a subclass of superrenormalizable interactions with ßi > O. For
instance, the interaction of electrons with photons in quantum electrodynamics (QED), -e1{;j'ljJ,
is renormalizable since according to eq. (2.1) one has ß = 4 - 0 - 3/2 x 2 - 1 = 0, whereas the
electron mass term -m1j;'ljJ is an example of a superrenormalizable interaction. Obviously, only
a finite number of renormalizable theories exists since ßi is bounded from above. It has been
shown that the theory is renormalizable (i. e. that ultraviolet divergences arising in calculations
of observables cancel as so on as all parameters2 of the theory are expressed in terms of physical
or renormalized quantities) if all interactions are renormalizable, see e.g. [133].
For quite a long time it was commonly believed that renormalizability is a fundamental principIe.
Indeed, a non-renormalizable quantum field theory is not well-defined if a finite number of
interactions enters the corresponding Lagrangian. Although one can always perform tree-Ievel
calculations, quantum corrections will produce ultraviolet divergences that cannot be absorbed in
are-definition of the parameters in the Lagrangian. In fact, renormalization can also be carried
out in the case of non-renormalizable theories if one includes in the Lagrangian all interactions
(an infinite number), that are consistent with symmetry principles of the corresponding theory
[133]. Such symmetry principles restrict, of course, the structure of possible interactions, but
1 In this thesis we will always use "natural" units with 1i = c = l.
2There is always a finite number of parameters in such renormalizable theories.
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