The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.1. What is eHective 13
and Heisenberg proposed to use an effective Lagrangian for calculating photon-photon scattering
at energies much smaller than the electron mass [134], [135], [136]
where F/LV = (-1/2) EJtvpu Fpu is the dual of the field strength tensor FJtv The ellipsis represent
further corrections ofhigher order in 1/me and a. We learn from the effective Lagrangian (2.3) that
photon-photon scattering at second order in a resulting from the process, which includes creation
and destruction of electron-positron pairs, can be represented at very low energies by simple fourphoton
contact interactions without knowing anything about the structure of photon-electron
interaction. This is because one cannot probe details of photon-electron coupling at energies
much smaller than the electron mass. In fact, the effective Lagrangian (2.3) is quite general, since
it includes all possible terms with four field strength tensors and without additional derivatives,
which are allowed by Lorentz and gauge symmetry. All information about the structure of photonelectron
interactions in QED is hidden in the values of the coupling constants of the (F,wFJtV) 2
and (F'WFJtV) 2 contact interactions.4 In QED one can derive these perturbatively to all orders of
a. If the fundamental theory (QED) would be not known, these coefficients could be fixed from
experiment. This simple example illustrates one of the basic principles of the effective field theory
method: the dynamics of a system at low energies is not sensitive to details of the interaction at
high energies.
The method of effective Lagrangians allows to calculate not only the leading low-energy approximation
of the scattering amplitude but also to systematically account for corrections. In the
case of photon-photon interactions this corrections are of two forms. First, even at order a 2
there are additional interactions containing at most four field strength tensors FJtv, FJtv and any
positive number of derivatives,5 like F Jtv (O/m�)FJtv FQßFQß , FJtv (0 2 /m�)FJtv FQßFQß , . . . . Secondly,
there are vertices at order a > 2 with and without derivatives. Some of them have the
same structure as the terms at order a = 2 but also many new interactions like, for instance,
1/m�(F Jtv FJtV)3 with more photons appear. Apart from various tree corrections loop diagrams
will contribute at higher orders. At each order in the low-momentum expansion one can absorb
all ultraviolet divergences arising from loops by corresponding counter terms in the effective Lagrangian.
Thus, calculations with an arbitrary accuracy can be performed within such an effective
theory. For one-loop calculations with the effective Lagrangian of Euler et al., see ref. [138].
In the above example for scattering of light by light the use of effective Lagrangians might be
useful, since it allows to simplify calculations at low-energies. For much more complicated full
QED calculation of the same process see the reference [139]. In some cases calculations within
an underlying fundamental theory are not possible: either the theory is not known or standard
perturbative methods are not available. This happens, in particular, far QCD at low energies. In
such situations effective field theories provide a simple and powerful way to perform a systematic
analysis of low-energy phenomena.
Effective field theory is not only an extremely useful tool for performing practical calculations,
which is alternative to standard methods of renarmalizable field theories. Presumably, it is the
only kind of physical theories that we have at present. Indeed, there is no evidence that the
4 Such a situation, when the strengths of couplings is controlled by some (heavy) resonances, happens quite often
in various effective theories. This is called resonance saturation. For some particular examples see ref. [137)
5 There are no such terms with more than four field strength tensors at order Q2 since in this case one would
need to couple at least six electrons, which requires three powers of Q (or more).