Nonrelativistic effective field theories

For a stable massive field in an appropriate particle-number sector, factor out the rapid phase associated with its rest mass. Modes corresponding to antiparticles or pair creation are then separated by an energy of order the mass and can be matched into local coefficients when inaccessible. The remaining field varies slowly. Its spatial momentum scales as mv and its residual energy as mv squared, with v much smaller than one. Time derivatives and spatial derivatives therefore have different velocity orders.

The leading kinetic equation is first order in time and second order in space. Expanding the relativistic dispersion gives a positive quadratic kinetic energy and a negative quartic momentum correction. The latter is a small perturbation when p/m is small; treating a truncated negative quartic as a globally valid Hamiltonian at arbitrarily large p would extend it beyond its purpose. Coupling to gauge fields requires gauge-covariant derivatives and matching additional spin or magnetic interactions rather than only replacing an energy formula.

For elastic S-wave scattering below inelastic thresholds, the effective-range expansion expresses momentum times cotangent of the phase shift as a series in momentum squared. The scattering length a and effective range re have length units. The amplitude has an additional imaginary term fixed by elastic unitarity. If a is comparable to the interaction range R, a perturbative low-momentum expansion may be suitable. A much larger scattering length introduces a new low-energy scale, inverse absolute a, below inverse R.

When k times a is order one or larger while kR remains small, repeatedly inserting the leading contact interaction is not suppressed. Summing those iterations can produce the shallow pole or enhanced scattering required by the data. A reorganized counting treats the enhanced contribution at leading order while still expanding range corrections systematically. This is an EFT with a controlled low-energy expansion, even though its leading contact sector is nonperturbative.

Pionless nuclear EFT, nonrelativistic atomic EFT, and nonrelativistic gauge theories retain different degrees of freedom and interactions. Long-range photons, explicit pions, or several closely spaced thresholds can introduce scales beyond the simple contact example. A shallow bound state can require a special leading treatment even when the underlying couplings are not weak. It is therefore unsafe to infer a universal diagram hierarchy solely from the canonical dimension of a contact operator.

Regularization and subtraction should make the chosen counting transparent without changing physical scattering. Different schemes reshuffle loop contributions and contact coefficients, so a large coefficient can partly reflect the chosen convention. Physical low-energy scales, the pole position, and observable convergence supply the stronger diagnostics. Relativistic corrections and range corrections can be independent expansion parameters; satisfying p/m much smaller than one does not by itself establish that a particular effective-range truncation meets a requested precision.