The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

14 2. Low-momentum effective theories for two nucleons
fundamental theories like QED or the Standard Model are really fundamental in the sense that they
remain valid for arbitrary high energies. Rather, these renormalizable theories may be low-energy
approximations of some other and more fundamental underlying theories. As already pointed
out before, non-renormalizable interactions are usually suppressed by inverse powers of energy,
at which new physics appear. For example, the interactions in the effective QED Lagrangian
(2.3) are suppressed by four powers of the inverse electron mass. In principle, there are two
ways of testing new physical effects beyond the Standard Model: either one can try to perform
more precise measurements of observables to test effects of non-renormalizable interactions or one
can increase the energy. At present, no experimental inconsistencies of the Standard Model to
data have been observed. This indicates that the energies at which new physics might become
significant are quite high and, probably, in the TeV region [140].
Up to now we have only considered effective field theories. The discussed ideas find also many
interesting applications in quantum mechanics and in other fields of physics. For example, the
standard multipole expansion of a complicated current source J(r, t) of size d, which generates
radiation with large wavelengths A » d, already uses the basic idea of effective theories: large
distance physics should be not very sensitive to details of the structure at short distances. That
is why one usually observes quick convergence of such multipole expansions.
In the next sections of this chapter we will concentrate, basically, on purely quantum mechanical
problems. All ideas and methods of effective theories, that we have discussed in the context of
relativistic quantum field theories, remain, of course, valid in this case. In particular, as already
stressed above, the low-energy behavior of a theory is not sensitive to details of the short-distance
interaction. 6 Following Lepage's lecture [141], we would like to enumerate here the basic steps of
constructing the effective theory:
• First, one should correctly incorporate the low-energy physics into the effective theory. This
must be known from the underlying theory. Note that if the underlying interaction has a
finite range, one can always restrict an effective theory to very low energies, at which the
details of the interaction are not important.7 In such a case one does not need to know
anything about the underlying theory.
• Secondly, one should introduce an ultraviolet cut-off to exclude high-moment um states. The
cut-off furthermore provides finiteness of various integrals arising by performing calculations.
• Finally, the missed short-range physics is represented in form of local correction terms with
arbitrary coefficients to the effective Hamiltonian. These coefficients have to be fixed from
the data. It is important that one should keep all local terms allowed by the symmetries of the
underlying theory. Putting new correction terms systematically removes the dependence of
the low-energy observables on the cut-off, which is compensated by "running" of parameters
in the effective Hamiltonian. Any given precision can be achieved with a finite number of
correction terms.
60ne needs, in general, higher energies to probe shorter-distance physics. This follows immediately from Heisenberg's
uncertainty relation.
7 A counter example is the Coulomb potential which has an infinite range. In the language of quantum field
theory, the Coulomb force results from the exchange of photons. Since photons are massless, they can not be
completely integrated out even at very low energies. One can, however, separate photons into "soft" and "hard"
on es and keep only "soft" photons explicitly within an effective theory.