The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

12 2. Low-momentum effective theories for two nucleons
these constrain in the same way the ultraviolet divergences. This makes it possible to absorb all
ultraviolet divergences arising from loops in parameters of the Lagrangian and the theory becomes
well-defined.
At first sight, such non-renormalizable field theories seem to be completely useless, because an
infinite number of vertices in the Lagrangian would lead to an infinite set of possible diagrams,
which contribute to a definite process. In such a case, practical calculations would be, of course, not
possible. However, as we will see below, in some cases in the low-energy limit non-renormalizable
interactions become weak and are strongly suppressed compared to renormalizable ones. This is
because coupling constants of non-renormalizable interactions can be expressed like Ci / An, w here
A is so me mass scale, n > 0, and Ci is so me dimensionless number. Thus, if no additional mass
scale appears in the corresponding theory or if all mass sc ales are much smaller than A, one
observes in the low-energy limit E ---+ 0 a strong suppression of non-renormalizable interactions
due to factors E / A. 3 So, in spite of infinitely many parameters entering the Langangian, the
theory possesses a predictive power. Indeed, if calculations are performed to a certain finite level
of accuracy, only a finite number of interactions is relevant. As so on as all relevant couplings are
fixed from so me processes, predictions can be made for other processes and observables. This is
w hat is usually understood under effective field theories (EFTs).
Figure 2.1: The leading l/m! contribution to photon-photon scattering at
second order in the fine structure constant CI' can be represented by a contact
interaction (filled cirele).
There are many physical situations, in which EFTs provide a very useful and sometimes the only
available way of performing practical calculations. If the fundamental theory is known, EFT can
be useful as its low-energy approximation. For example, the leading photon-photon scattering
diagram in QED, shown in fig. 2.1, can be calculated from the QED Lagrangian at one loop order
where 'I/J denotes the electron fields, Fftv == 8ft AV - 8V Aft is the field strength tensor, me and
CI' = e 2 /(47r) rv 1/137 denote the electron mass and electromagnetic fine structure constant,
respectively. In eq. (2.2) we have not shown gauge fixing terms. Already in 1936 Euler, Kockel
3 This statement obviously holds at the tree-level. Quantum corrections include loop integrals, which may
be ultraviolet divergent and should be regularized and renormalized. This requires more careful considerations.
Weinberg has shown that quantum corrections are also suppressed [60).
(2.2)