# nLab effective quantum field theory

under construction

# Contents

## Idea

While fundamental physics is at some level well described by quantum field theory, a typical Lagrangian used to define such a QFT can reasonably be expected to define only degrees of freedom and interactions that are relevant up to some given energy scale. In this perspective one speaks of the theory as being the effective quantum field theory of some – possibly known but possibly unspecified – more fundamental theory.

An example (historically the first to be successfully considered) is the Fermi theory of beta decay of hadrons: this contains interactions of four fermions at a time, for instance a process in which a neutron decays into a collection consisting of a proton, an electron and a neutrino. Later it was discovered that, more fundamentally, this is not a single reaction but is composed out of several other interactions that involve exchanges of W-bosons between these four particles. Nevertheless, Fermi’s original effective theory made very precise predictions at energy scales less than 10 MeV?. The reason is that the $W$-boson has mass several orders of magnitude higher than that (about 80 GeV) and was thus effectively invisible at these low energies.

The low energy expansion of any unitary, relativistic, crossing symmetric? S-matrix can be described by an effective quantum field theory.

In the perspective of effective field theory notably unrenormalizable Lagrangians can still make perfect sense as effective theories and give rise to well defined predictions: they can be effective approximations to renormalizable or even degreewise finite more fundamental theories. This is sometimes called a UV completion of the given effective theory.

For instance gravity – which is notoriously non-renormalizable – makes perfect sense as an effective field theory (see for instance the introduction in (Donoghue). It is in principle possible that there is some more fundamental theory with plenty of excitations at high energies that is however degreewise finite in perturbation theory, whose effective description at low energy is given by the unrenormalizable Einstein-Hilbert action. (For instance, string theory is meant to be such a theory.)

## Theory

The technique of effective field theory is based on the following fact:

###### Proposition

For a given set of asymptotic states, perturbation theory with the most general Lagrangian containing all terms allowed by the assumed symmetries will yield the most general S-matrix elements consistent with analyticity, perturbative unitarity?, cluster decomposition and the assumed symmetries.

This is due to (Weinberg 1979) and (Leutweyler94).

Based on this fact, one obtains an effective approximation to a given more fundamental theory (which may or may not be actually known) by

1. choosing the (sub)set of fields to be considered;

2. writing down a Lagrangian

$L_{eff} = \sum_i c_i O_i$

that contains all the possible polynomial interaction terms $O_i$ of these fields scaled by their expected/known energy scale $[O_i] = d_i$, up to a maximal energy scale

(this will in general contain lots of direct interaction that in the fundamental theory are really compound interactions)

with $c_i \propto \frac{1}{\Lambda^{d_i - dim X}}$;

3. finally one fixes all the coupling constants of all these interactions by

• either deriving them from a known fundamental theory by integrating out higher energy effects in that theory;

• or, otherwise, measuring them in the laboratory. The point being that due to the energy cutoff, this is guaranteed to be a finite number of parameters. After these have been determined, all remaining quantities given by the Lagrangian are then predictions of the effective theory.

## Examples

### Chiral perturbation theory

chiral perturbation theory? is an effective approximation of QCD in the light quark sector.

(…)

### Neutrino masses (?)

On neutrino masses and the standard model of particle physics as an effective field theory:

I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor $M^{-1}$, and that therefore with a reasonable estimate of $M$ could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of $10^{16} GeV$. (Weinberg 09, p. 15)

### String theory and gravity coupled to gauge theory

The string scattering amplitudes for superstrings are finite (fully proven so for low loop order and with various plausibility arguments for higher loop order, see at string scattering amplitudes for more), hence define a UV-complete S-matrix. The corresponding low energy effective field theories are theories of supergravity coupled to gauge theory. (type II supergravity, heterotic supergravity).

## References

The modern picture of effective low-energy QFT goes back to

• L. P. Kadanoff, Scaling laws for Ising models near $T_c$ , Physica 2 (1966);

• Kenneth Wilson, Renormalization group and critical phenomena , I., Physical review B 4(9) (1971).

• Joe Polchinski, Renormalization and effective Lagrangians , Nuclear Phys. B B231 (1984).

• Steven Weinberg, Physica 96 A (1979) 327

• H. Leutwyler, Ann. Phys., NY 235 (1994) 165.

A review is in

A brief introduction aimed at mathematicians is in

A standard textbook adopting this perspective is

• Steven Weinberg The Quantum Theory of Fields (Cambridge University Press,Cambridge,1995).

whose author describes his goal as: “This is intended to be a book on quantum field theory for the era of effective field theory.” Another book which takes the effective-field-theory approach to QFT is

• Anthony Zee, Quantum Field Theory in a Nutshell (Princeton University Press, second edition, 2010).

Introductory lecture notes are for instance in

The point that perturbatively non-renormalizable theories may be regarded as effective field theories at each energy scale was highligted in

• J. Gomis and Steven Weinberg, Are nonrenormalizable gauge theories renormalizable?, Nucl. Phys. B 469 (1996) 473.

Notably the theory of gravity based on the standard Einstein-Hilbert action may be regarded as just an effective QFT, which makes some of its notorious problems be non-problems:

and in the context of perturbation theory in AQFT:

Comments on this point are also in

Revised on January 20, 2015 21:31:35 by Urs Schreiber (88.100.66.95)