The absence of such a framework would indeed be a crisis for theoretical physics, since real theoretical predictions are necessarily approximate. Controllable results always require some understanding of the size of the contributions being neglected in any given calculation. If quantum effects in general relativity cannot be quantified, this must undermine our satisfaction with the experimental success of its classical predictions.
It is the purpose of this article to present the modern point of view on these issues, which has emerged since the early 1980's. According to this point of view there is no such crisis, because the problems of quantizing gravity within the experimentally accessible situations are similar to those which arise in a host of other nongravitational applications throughout physics. As such, the size of quantum corrections can be safely estimated and are extremely small. The theoretical framework which allows this quantification is the formalism of
effective field theories, whose explanation makes up the better part of this article. In so doing we shall see that although there can be little doubt of the final outcome, the explicit determination of the size of subleading quantum effects in gravity has in many cases come only relatively recently, and a complete quantitative analysis of the size of quantum corrections remains a work in progress.
1.2 Identifying where the problems lie
This is not to say that there are no challenging problems remaining in reconciling quantum mechanics with gravity. On the contrary, many of the most interesting issues remain to be solved, including the identification of what the right observables should be, and understanding how space and time might emerge from more microscopic considerations. For the rest of the discussion it is useful to separate these deep, unsolved issues of principle from the more prosaic, technical problem of general relativity's nonrenormalizability.
There have been a number of heroic attempts to quantize gravity along the lines of other field theories [
80,
54,
9,
70,
137,
53,
69,
112,
50,
?,
52,
25,
38,
10,
11,
12,
6,
16,
8,
7,
13,
14,
15,
18]
, and it was recognized early on that general relativity is not renormalizable. It is this technical problem of nonrenormalizability which in practice has been the obstruction to performing quantum calculations with general relativity.
As usually stated, the difficulty with nonrenormalizable theories is that they are not predictive, since the obtention of welldefined predictions potentially requires an infinite number of divergent renormalizations.
It is not the main point of the present review to recap the techniques used when quantizing the gravitational field, nor to describe in detail its renormalizability. Rather, this review is intended to describe the modern picture of what renormalization means, and why nonrenormalizable theories need not preclude making meaningful predictions. This point of view is now wellestablished in many areas – such as particle, nuclear, and condensedmatter physics – where nonrenormalizable theories arise. In these other areas of physics predictions can be made with nonrenormalizable theories (including quantum corrections) and the resulting predictions are wellverified experimentally.
The key to making these predictions is to recognize that they must be made within the context of a lowenergy expansion, in powers of
$E/M$
(energy divided by some heavy scale intrinsic to the problem). Within the validity of this expansion theoretical predictions are under complete control.
The lesson for quantum gravity is clear: Nonrenormalizability is not in itself an obstruction to performing predictive quantum calculations, provided the lowenergy nature of these predictions in powers of
$E/M$
, for some
$M$
, is borne in mind. What plays the role of the heavy scale
$M$
in the case of quantum gravity? It is tempting to identify this scale with the Planck mass
${M}_{p}$
, where
${M}_{p}^{2}=8\pi G$
(with
$G$
denoting Newton's constant), and in some circumstances this is the right choice. But as we shall see
$M$
need not be
${M}_{p}$
, and for some applications might instead be the electron mass
${m}_{e}$
, or some other scale. One of the points of quantifying the size of quantum corrections is to identify more precisely what the important scales are for a given quantumgravity application.
Once it is understood how to use nonrenormalizable theories, the size of quantum effects can be quantified, and it becomes clear where the real problems of quantum gravity are pressing and where they are not. In particular, the lowenergy expansion proves to be an extremely good approximation for all of the present experimental tests of gravity, making quantum corrections negligible for these tests. By contrast, the lowenergy nature of quantumgravity predictions implies that quantum effects are important where gravitational fields become very strong, such as inside black holes or near cosmological singularities. This is what makes the study of these situations so interesting: it is through their study that progress on the more fundamental issues of quantum gravity is likely to come.
1.3 A road map
The remainder of this article is organized in the following way:
Section
2 about effective field theories does not involve gravity at all, but instead first describes why effective field theories are useful in other branches of physics. The discussion is kept concrete by considering a simple toy model, for which it is argued how some applications make it useful to keep track of how small ratios of energy scales appear in physical observables. In particular, considerable simplification can be achieved if an expansion in small energy ratios is performed as early as possible in the calculation of lowenergy observables. The theoretical tool for achieving this simplification is the effective Lagrangian, and its definition and use is briefly summarized using the toy model as an explicit example.
Section
3 , which deals with quantum gravity as an effective theory, describes how the tools of the previous section may be applied to calculating quantum effects including the gravitational field.
In particular, it is shown how to make predictions despite general relativity's nonrenormalizability, since effective Lagrangians are generically not renormalizable. As we shall see, however, some of the main results one would like to have regarding the size of quantum corrections to arbitrary loop orders remain incomplete.
In Section
4 explicit applications of these ideas are described in this section, which use the above results to compute quantum corrections to several gravitational results for two kinds of sources. These calculations compute the leading quantum corrections to Newton's Law between two slowlymoving point particles, and to the gravitational force between two cosmic strings (both in
$3+1$
spacetime dimensions).
In the final Section
5 conclusions are briefly summarized.
2 Effective Field Theories
This section describes the effectiveLagrangian technique within the context of a simple toy model, closely following the discussion of [
27]
.
In all branches of theoretical physics a key part of any good prediction is a careful assessment of the theoretical error which the prediction carries. Such an assessment is a precondition for any detailed quantitative comparison with experiment. As is clear from numerous examples throughout physics, this assessment of error usually is reliably determined based on an understanding of the small quantities which control the corrections to the approximations used when making predictions.
Perhaps the most famous example of such a small quantity might be the finestructure constant,
$\alpha ={e}^{2}/\u0127c$
, which quantifies the corrections to electromagnetic predictions of elementary particle properties or atomic energy levels.
2.1 The utility of lowenergy approximations
It sometimes happens that predictions are much more accurate than would be expected based on an assessment of the approximations on which they appear to be based. A famous example of this is encountered in the precision tests of quantum electrodynamics, where the value of the finestructure constant,
$\alpha $
, was until recently obtained using the Josephson effect in superconductivity.
A DC potential difference applied at the boundary between two superconductors can produce an AC Josephson current whose frequency is precisely related to the size of the applied potential and the electron's charge. Precision measurements of frequency and voltage are in this way converted into a precise measurement of
$e/\u0127$
, and so of
$\alpha $
. But use of this effect to determine
$\alpha $
only makes sense if the predicted relationship between frequency and voltage is also known to an accuracy which is better than the uncertainty in
$\alpha $
.
It is, at first sight, puzzling how such an accurate prediction for this effect can be possible.
After all, the prediction is made within the BCS theory of superconductivity (see, for example, [
138]
), which ignores most of the mutual interactions of electrons, focussing instead on a particular pairing interaction due to phonon exchange. Radical though this approximation might appear to be, the theory works rather well (in fact, surprisingly well), with its predictions often agreeing with experiment to within several percent. But expecting successful predictions with an accuracy of parts per million or better would appear to be optimistic indeed!
The astounding theoretical accuracy required to successfully predict the Josephson frequency may be understood at another level, however. The key observation is that this prediction does not rely at all on the details of the BCS theory, depending instead only on the symmetrybreaking pattern which it predicts. Once it is known that a superconductor spontaneously breaks the
$U\left(1\right)$
gauge symmetry of electromagnetism, the Josephson prediction follows on general grounds in the lowenergy limit (for a discussion of superconductors in an effectiveLagrangian spirit aimed at a particlephysics audience see [
150]
). The validity of the prediction is therefore not controlled by the approximations made in the BCS theory, since any theory with the same lowenergy symmetrybreaking pattern shares the same predictions.
The accuracy of the predictions for the Josephson effect are therefore founded on symmetry arguments, and on the validity of a lowenergy approximation. Quantitatively, the lowenergy approximation involves the neglect of powers of the ratio of two scales,
$\omega /\Omega $
, where
$\omega $
is the low energy scale of the observable under consideration – like the applied voltage in the Josephson effect – and
$\Omega $
is the higher energy scale – such as the superconducting gap energy – which is intrinsic to the system under study.
Indeed, arguments based on a similar lowenergy approximation may also be used to explain the surprising accuracy of many other successful models throughout physics, including the BCS theory itself [
129,
135,
134,
35]
. This is accomplished by showing that only the specific interactions used by the BCS theory are relevant at low energies, with all others being suppressed in their effects by powers of a small energy ratio.
Although many of these arguments were undoubtedly known in various forms by the experts in various fields since very early days, the systematic development of these arguments into precision calculational techniques has happened more recently. With this development has come considerable crossfertilization of techniques between disciplines, with the realization that the same methods play a role across diverse disciplines within physics.
The remainder of this article briefly summarizes the techniques which have been developed to exploit lowenergy approximations. These are most efficiently expressed using effectiveLagrangian methods, which are designed to take advantage of the simplicity of the lowenergy limit as early as possible within a calculation. The gain in simplicity so obtained can be the decisive difference between a calculation's being feasible rather than being too difficult to entertain.
Besides providing this kind of practical advantage, effectiveLagrangian techniques also bring real conceptual benefits because of the clear separation they permit between of the effects of
different scales. Both of these kinds of advantages are illustrated here using explicit examples.
First Section
2.2 presents a toy model involving two spinless particles to illustrate the general method, as well as some of its calculational advantages. This is followed by a short discussion of the conceptual advantages, with quantum corrections to classical general relativity, and the associated problem of the nonrenormalizability of gravity, taken as the illustrative example.
2.2 A toy example
In order to make the discussion as concrete as possible, consider the following model for a single complex scalar field
$\phi $
:
$$\begin{array}{c}\mathcal{\mathcal{L}}={\partial}_{\mu}{\phi}^{*}{\partial}^{\mu}\phi V\left({\phi}^{*}\phi \right),\end{array}$$ 
(1)

with
$$\begin{array}{c}V=\frac{{\lambda}^{2}}{4}{\left({\phi}^{*}\phi {v}^{2}\right)}^{2}.\end{array}$$ 
(2)

This theory enjoys a continuous
$U\left(1\right)$
symmetry of the form
$\phi \to {e}^{i\omega}\phi $
, where the parameter
$\omega $
is a constant. The two parameters of the model are
$\lambda $
and
$v$
. Since
$v$
is the only dimensionful quantity it sets the model's overall energy scale.
The semiclassical approximation is justified if the dimensionless quantity
$\lambda $
should be sufficiently small. In this approximation the vacuum field configuration is found by minimizing the system's energy density, and so is given (up to a
$U\left(1\right)$
transformation) by
$\phi =v$
. For small
$\lambda $
the spectrum consists of two weaklyinteracting particle types described by the fields
$\mathcal{\mathcal{R}}$
and
$\mathcal{\mathcal{I}}$
, where
$\phi =\left(v+\frac{1}{\sqrt{2}}\mathcal{\mathcal{R}}\right)+\frac{i}{\sqrt{2}}\mathcal{\mathcal{I}}$
. To leading order in
$\lambda $
the particle masses are
${m}_{\mathcal{\mathcal{I}}}=0$
and
${m}_{\mathcal{\mathcal{R}}}=\lambda v$
.
The lowenergy regime in this model is
$E\ll {m}_{\mathcal{\mathcal{R}}}$
. The masslessness of
$\mathcal{\mathcal{I}}$
ensures the existence of degrees of freedom in this regime, with the potential for nontrivial lowenergy interactions, which we next explore.
2.2.1 Masslessparticle scattering
The interactions amongst the particles in this model are given by the scalar potential:
$$\begin{array}{c}V=\frac{{\lambda}^{2}}{16}{\left(2\sqrt{2}v\mathcal{\mathcal{R}}+{\mathcal{\mathcal{R}}}^{2}+{\mathcal{\mathcal{I}}}^{2}\right)}^{2}.\end{array}$$ 
(3)

Figure 1
: The Feynman graphs responsible for treelevel
$\mathcal{\mathcal{R}}\text{\u2013}\mathcal{\mathcal{I}}$
scattering in the toy model. Here solid lines denote
$\mathcal{\mathcal{R}}$
particles and dashed lines represent
$\mathcal{\mathcal{I}}$
particles.
Imagine using the potential of Equation ( 3 ) to calculate the amplitude for the scattering of
$\mathcal{\mathcal{I}}$
particles at low energies to lowestorder in
$\lambda $
. For example, the Feynman graphs describing treelevel
$\mathcal{\mathcal{I}}$
–
$\mathcal{\mathcal{R}}$
scattering are given in Figure 1 . The
$S$
matrix obtained by evaluating the analogous treelevel diagrams for
$\mathcal{\mathcal{I}}$
selfscattering is proportional to the following invariant amplitude:
$$\begin{array}{c}\mathcal{A}=\frac{3{\lambda}^{2}}{2}+{\left(\frac{{\lambda}^{2}v}{\sqrt{2}}\right)}^{2}\left[\frac{1}{(s+r{)}^{2}+{m}_{\mathcal{\mathcal{R}}}^{2}i\epsilon}+\frac{1}{(r{r}^{\prime}{)}^{2}+{m}_{\mathcal{\mathcal{R}}}^{2}i\epsilon}+\frac{1}{(r{s}^{\prime}{)}^{2}+{m}_{\mathcal{\mathcal{R}}}^{2}i\epsilon}\right],\end{array}$$ 
(4)

where
${s}^{\mu}$
and
${r}^{\mu}$
(and
${{s}^{\prime}}^{\mu}$
and
${{r}^{\prime}}^{\mu}$
) are the 4momenta of the initial (and final) particles.
An interesting feature of this amplitude is that when it is expanded in powers of external fourmomenta, both its leading and nexttoleading terms vanish. That is
$$\begin{array}{ccc}\mathcal{A}& =& \left[\frac{3{\lambda}^{2}}{2}+\frac{3}{{m}_{\mathcal{\mathcal{R}}}^{2}}{\left(\frac{{\lambda}^{2}v}{\sqrt{2}}\right)}^{2}\right]+\frac{2}{{m}_{\mathcal{\mathcal{R}}}^{4}}{\left(\frac{{\lambda}^{2}v}{\sqrt{2}}\right)}^{2}\left[r\cdot s+r\cdot {r}^{\prime}+r\cdot {s}^{\prime}\right]+\mathcal{O}(quarticinmomenta)\end{array}$$  
$$\begin{array}{ccc}& =& 0+\mathcal{O}(quarticinmomenta).\end{array}$$ 
(5)

The last equality uses conservation of 4momentum,
${s}^{\mu}+{r}^{\mu}={{s}^{\prime}}^{\mu}+{{r}^{\prime}}^{\mu}$
, and the massless massshell condition
${r}^{2}=0$
. Something similar occurs for
$\mathcal{\mathcal{R}}$
–
$\mathcal{\mathcal{I}}$
scattering, which also vanishes due to a cancellation amongst the graphs of Figure 1 in the zeromomentum limit.
Clearly the lowenergy particles interact more weakly than would be expected given a cursory inspection of the scalar potential, Equation (
3 ), since at tree level the lowenergy scattering rate is suppressed by at least eight powers of the small energy ratio
$r=E/{m}_{R}$
. The real size of the scattering rate might depend crucially on the relative size of
$r$
and
${\lambda}^{2}$
, should the vanishing of the leading lowenergy terms turn out to be an artifact of leadingorder perturbation theory.
If
$\mathcal{\mathcal{I}}$
scattering were of direct experimental interest, one can imagine considerable effort being invested in obtaining higherorder corrections to this lowenergy result. And the final result proves to be quite interesting: As may be verified by explicit calculation, the first two terms in the lowenergy expansion of
$\mathcal{A}$
vanish orderbyorder in perturbation theory. Furthermore, a similar suppression turns out also to hold for all other amplitudes involving
$\mathcal{\mathcal{I}}$
particles, with the
$n$
point amplitude for
$\mathcal{\mathcal{I}}$
scattering being suppressed by
$n$
powers of
$r$
.
Clearly the hard way to understand these lowenergy results is to first compute to all orders in
$\lambda $
and then expand the result in powers of
$r$
. A much more efficient approach exploits the simplicity of small
$r$
before calculating scattering amplitudes.
2.3 The toy model revisited
The key to understanding this model's lowenergy limit is to recognize that the lowenergy suppression of scattering amplitudes (as well as the exact masslessness of the light particle) is a consequence of the theory's
$U\left(1\right)$
symmetry. (The massless state has these properties because it is this symmetry's Nambu–Goldstone boson. The earliest general formulation of nonAbelian Goldstoneboson interactions arose through the study of lowenergy pion interactions [
144,
145,
32,
72]
; for reviews of Goldstone boson properties see [
83,
106]
; see also [
147,
28]
.) The simplicity of the lowenergy behaviour is therefore best displayed by

∙
making the symmetry explicit for the lowenergy degrees of freedom, and

∙
performing the lowenergy approximation as early as possible.
2.3.1 Exhibiting the symmetry
The
$U\left(1\right)$
symmetry can be made to act exclusively on the field which represents the light particle by parameterizing the theory using a different set of variables than
$\mathcal{\mathcal{I}}$
and
$\mathcal{\mathcal{R}}$
. To this end imagine instead using polar coordinates in field space
$$\begin{array}{c}\phi \left(x\right)=\chi \left(x\right){e}^{i\theta \left(x\right)}.\end{array}$$ 
(6)

In terms of
$\theta $
and
$\chi $
the action of the
$U\left(1\right)$
symmetry is simply
$\theta \to \theta +\omega $
, and the model's Lagrangian becomes
$$\begin{array}{c}\mathcal{\mathcal{L}}={\partial}_{\mu}\chi {\partial}^{\mu}\chi {\chi}^{2}{\partial}_{\mu}\theta {\partial}^{\mu}\theta V\left({\chi}^{2}\right).\end{array}$$ 
(7)

The semiclassical spectrum of this theory is found by expanding
$\mathcal{\mathcal{L}}$
in powers of the canonicallynormalized fluctuations,
${\chi}^{\prime}=\sqrt{2}(\chi v)$
and
${\theta}^{\prime}=\sqrt{2}v\theta $
, about the vacuum
$\chi =v$
, revealing that
${\chi}^{\prime}$
describes the mass
${m}_{R}$
particle while
${\theta}^{\prime}$
represents the massless particle.
With the
$U\left(1\right)$
symmetry realized purely on the massless field
$\theta $
, we may expect good things to happen if we identify the lowenergy dynamics.
2.3.2 Timely performance the lowenergy approximation
To properly exploit the symmetry of the lowenergy limit we integrate out all of the highenergy degrees of freedom as the very first step, leaving the inclusion of the lowenergy degrees of freedom to last. This is done most efficiently by computing the following lowenergy effective (or Wilson) action.
A conceptually simple (but cumbersome in practice) way to split degrees of freedom into `heavy' and `light' categories is to classify all field modes in momentum space as heavy if (in Euclidean signature) they satisfy
${p}^{2}+{m}^{2}>{\Lambda}^{2}$
,where
$m$
is the corresponding particle mass and
$\Lambda $
is an appropriately chosen cutoff.
Light modes are then all of those which are not heavy. The cutoff
$\Lambda $
, which defines the boundary between these two kinds of modes,is chosen to lie well below the highenergy scale (i.e., well below
${m}_{R}$
in the toy model),but is also chosen to lie well above the lowenergy scale of ultimate interest (like the centreofmass energies
$E$
of lowenergy scattering amplitudes). Notice that in the toy model the heavy degrees of freedom defined by this split include all modes of the field
${\chi}^{\prime}$
, as well as the highfrequency components of the massless field
${\theta}^{\prime}$
.
If
$h$
and
$\ell $
schematically denote the fields which are, respectively, heavy or light in this characterization, then the influence of heavy fields on lightparticle scattering at low energies is completely encoded in the following effective Lagrangian:
$$\begin{array}{c}exp\left[i\int {d}^{4}x{\mathcal{\mathcal{L}}}_{eff}(\ell ,\Lambda )\right]=\int \mathcal{D}{h}_{\Lambda}exp\left[\int {d}^{4}x\mathcal{\mathcal{L}}(\ell ,h)\right].\end{array}$$ 
(8)

The
$\Lambda $
dependence which is introduced by the lowenergy/highenergy split of the integration measure is indicated explicitly in this equation.
Physical observables at low energies are now computed by performing the remaining path integral over the light degrees of freedom only. By virtue of its definition, each configuration in the integration over light fields is weighted by a factor of
$exp\left[i\int {d}^{4}x{\mathcal{\mathcal{L}}}_{eff}(\ell )\right]$
implying that the effective Lagrangian weights the lowenergy amplitudes in precisely the same way as the classical Lagrangian does for the integral over both heavy and light degrees of freedom. In detail, the effects of virtual contributions of heavy states appear within the lowenergy theory through the contributions of new effective interactions, such as are considered in detail for the toy model in some of the next sections (see, e.g., Sections 2.3.3 , 2.3.4 , and 2.5.2 ).
Although this kind of lowenergy/highenergy split in terms of cutoffs most simply illustrates the conceptual points of interest, in practical calculations it is usually dimensional regularization which is more useful. This is particularly true for theories (like general relativity) involving gauge symmetries, which can be conveniently kept manifest using dimensional regularization. We therefore return to this point in subsequent sections to explain how dimensional regularization can be used with an effective field theory.
2.3.3 Implications for the lowenergy limit
Now comes the main point. When applied to the toy model, the condition of symmetry and the restriction to the lowenergy limit together have strong implications for
${\mathcal{\mathcal{L}}}_{eff}\left(\theta \right)$
. Specifically:

∙
Invariance of
${\mathcal{\mathcal{L}}}_{eff}\left(\theta \right)$
under the symmetry
$\theta \to \theta +\omega $
implies
${\mathcal{\mathcal{L}}}_{eff}$
can depend on
$\theta $
only through the invariant quantity
${\partial}_{\mu}\theta $
.

∙
Interest in the lowenergy limit permits the expansion of
${\mathcal{\mathcal{L}}}_{eff}$
in powers of derivatives of
$\theta $
.
Because only lowenergy functional integrals remain to be performed, higher powers of
${\partial}_{\mu}\theta $
correspond in a calculable way to higher suppression of observables by powers of
$E/{m}_{\mathcal{\mathcal{R}}}$
.
Combining these two observations leads to the following form for
${\mathcal{\mathcal{L}}}_{eff}$
:
where the ellipses represent terms which involve more than six derivatives, and so more than two inverse powers of
${m}_{\mathcal{\mathcal{R}}}$
.
A straightforward calculation confirms the form (
9 ) in perturbation theory, but with the additional information
$$\begin{array}{c}{a}_{pert}=\frac{1}{4{\lambda}^{2}}+\mathcal{O}\left({\lambda}^{0}\right),{b}_{pert}=\frac{1}{4{\lambda}^{2}}+\mathcal{O}\left({\lambda}^{0}\right),{c}_{pert}=\frac{1}{4{\lambda}^{2}}+\mathcal{O}\left({\lambda}^{0}\right).\end{array}$$ 
(10)

In this formulation it is clear that each additional factor of
$\theta $
is always accompanied by a derivative, and so implies an additional power of
$r$
in its contribution to all lightparticle scattering amplitudes. Because Equation ( 9 ) is derived assuming only general properties of the low energy effective Lagrangian, its consequences (such as the suppression by
${r}^{n}$
of lowenergy
$n$
point amplitudes) are insensitive of the details of the underlying model. They apply, in particular, to all orders in
$\lambda $
.
Conversely, the details of the underlying physics only enter through specific predictions, such as Equations (
10 ), for the lowenergy coefficients
$a$
,
$b$
, and
$c$
. Different models having a
$U\left(1\right)$
Goldstone boson in their lowenergy spectrum can differ in the lowenergy selfinteractions of this particle only through the values they predict for these coefficients.
2.3.4 Redundant interactions
The effective Lagrangian ( 9 ) does not contain all possible polynomials of
${\partial}_{\mu}\theta $
. For example, two terms involving 4 derivatives which are not written are
$$\begin{array}{c}{\mathcal{\mathcal{L}}}_{redundant}=d\square \theta \square \theta +e{\partial}_{\mu}\theta \square {\partial}^{\mu}\theta ,\end{array}$$ 
(11)

where
$d$
and
$e$
are arbitrary real constants. These terms are omitted because their inclusion would not alter any of the predictions of
${\mathcal{\mathcal{L}}}_{eff}$
. Because of this, interactions such as those in Equation ( 11 ) are known as redundant interactions.
There are two reasons why such terms do not contribute to physical observables. The first reason is the old saw that states that total derivatives may be dropped from an action. More precisely, such terms may be integrated to give either topological contributions or surface terms evaluated at the system's boundary. They may therefore be dropped provided that none of the physics of interest depends on the topology or what happens on the system's boundaries. (See, however, [
2]
and references therein for a concrete example where boundary effects play an important role within an effective field theory.) Certainly boundary terms are irrelevant to the form of the classical field equations far from the boundary. They also do not contribute perturbatively to scattering amplitudes, as may be seen from the Feynman rules which are obtained from a simple total derivative interaction like
$$\begin{array}{c}\Delta \mathcal{\mathcal{L}}=g{\partial}_{\mu}({\partial}^{\mu}\theta \square \theta )=g\left(\square \theta \square \theta +{\partial}^{\mu}\theta \square {\partial}_{\mu}\theta \right),\end{array}$$ 
(12)

since these are proportional to
$$\begin{array}{c}g({p}^{2}{q}^{2}+{p}^{\mu}{q}_{\mu}{q}^{2}){\delta}^{4}(p+q)=g{q}^{2}{p}^{\mu}({p}_{\mu}+{q}_{\mu}){\delta}^{4}(p+q)=0.\end{array}$$ 
(13)

This shows that the two interactions of Equation ( 11 ) are not independent, since we can integrate by parts to replace the couplings
$(d,e)$
with
$({d}^{\prime},{e}^{\prime})=(de,0)$
.
The second reason why interactions might be physically irrelevant (and so redundant) is if they may be removed by performing a field redefinition. For instance under the infinitesimal redefinition
$\delta \theta =A\square \theta $
, the leading term in the lowenergy action transforms to
$$\begin{array}{c}\delta ({v}^{2}{\partial}_{\mu}\theta {\partial}^{\mu}\theta )=2A{v}^{2}{\partial}_{\mu}\theta \square {\partial}^{\mu}\theta .\end{array}$$ 
(14)

This redefinition can be used to set the effective coupling
$e$
to zero, simply by choosing
$2A{v}^{2}=e$
.
This argument can be repeated orderbyorder in powers of
$1/{m}_{R}$
to remove more and more terms in
${\mathcal{\mathcal{L}}}_{eff}$
without affecting physical observables.
Since the variation of the lowestorder action is always proportional to its equations of motion, it is possible to remove in this way
any interaction which vanishes when evaluated at the solution to the lowerorder equations of motion. Of course, a certain amount of care must be used when so doing. For instance, if our interest is in how the
$\theta $
field affects the interaction energy of classical sources, we must add a source coupling
$\Delta \mathcal{\mathcal{L}}={J}^{\mu}{\partial}_{\mu}\theta $
to the Lagrangian. Once this is done the lowestorder equations of motion become
$2{v}^{2}\square \theta ={\partial}_{\mu}{J}^{\mu}$
, and so an effective interaction like
$\square \theta \square \theta $
is no longer completely redundant. It is instead equivalent to the contact interactions like
$({\partial}_{\mu}{J}^{\mu}{)}^{2}/(4{v}^{4})$
.
2.4 Lessons learned
It is clear that the kind of discussion given for the toy model can be performed equally well for any other system having two wellseparated energy scales. There is a number of features of this example which also generalize to these other systems. It is the purpose of this section to briefly list some of these features.
2.4.1 Why are effective Lagrangians not more complicated?
${\mathcal{\mathcal{L}}}_{eff}$
as computed in the toy model is not a completely arbitrary functional of its argument
$\theta $
.
For example,
${\mathcal{\mathcal{L}}}_{eff}$
is real and not complex, and it is local in the sense that (to any finite order in
$1/{m}_{R}$
) it consists of a finite sum of powers of the field
$\theta $
and its derivatives, all evaluated at the same point.
Why should this be so? Both of these turn out to be general features (so long as only massive degrees of freedom are integrated out) which are inherited from properties of the underlying physics at higher energies:

Reality:
The reality of
${\mathcal{\mathcal{L}}}_{eff}$
is a consequence of the unitarity of the underlying theory, and the observation that the degrees of freedom which are integrated out to obtain
${\mathcal{\mathcal{L}}}_{eff}$
are excluded purely on the grounds of their energy. As a result, if no heavy degrees of freedom appear as part of an initial state, energy conservation precludes their being produced by scattering and so appearing in the final state.
Since
${\mathcal{\mathcal{L}}}_{eff}$
is constructed to reproduce this time evolution of the full theory, it must be real in order to give a Hermitian Hamiltonian as is required by unitary time evolution^{
$\text{1}$
}
.

Locality:
The locality of
${\mathcal{\mathcal{L}}}_{eff}$
is also a consequence of excluding highenergy states in its definition, together with the Heisenberg Uncertainty Relations. Although energy and momentum conservation preclude the direct production of heavy particles (like those described by
$\chi $
in the toy model) from an initial lowenergy particle configuration, it does not preclude their virtual production.
That is, heavy particles may be produced so long as they are then redestroyed sufficiently quickly. Such virtual production is possible because the Uncertainty Relations permit energy not to be precisely conserved for states which do not live indefinitely long. A virtual state whose production requires energy nonconservation of order
$\Delta E\sim M$
therefore cannot live longer than
$\Delta t\sim 1/M$
, and so its influence must appear as being local in time when observed only with probes having much smaller energy. Similar arguments imply locality in space for momentumconserving systems. (This is a heuristic explanation of what goes under the name operator product expansion [
156,
41]
in the quantum field theory literature.) Since it is the mass
$M$
of the heavy particle which sets the scale over which locality applies once it is integrated out, it is
$1/M$
which appears with derivatives of lowenergy fields when
${\mathcal{\mathcal{L}}}_{eff}$
is written in a derivative expansion.
2.5 Predictiveness and power counting
The entire rationale of an effective Lagrangian is to incorporate the virtual effects of highenergy particles in lowenergy processes, orderbyorder in powers of the small ratio
$r$
of these two scales (e.g.,
$r=E/{m}_{\mathcal{\mathcal{R}}}$
in the toy model). In order to use an effective Lagrangian it is therefore necessary to know which terms contribute to physical processes to any given order in
$r$
.
This determination is explicitly possible if the lowenergy degrees of freedom are weakly interacting, because in this case perturbation theory in the weak interactions may be analyzed graphically, permitting the use of powercounting arguments to systematically determine where powers of
$r$
originate. Notice that the assumption of a weaklyinteracting lowenergy theory does not presuppose the underlying physics to be also weakly interacting. For instance, for the toy model the Goldstone boson of the lowenergy theory is weakly interacting provided only that the
$U\left(1\right)$
symmetry is spontaneously broken, since its interactions are all suppressed by powers of
$r$
. Notice that this is true independent of the size of the coupling
$\lambda $
of the underlying theory.
For example, in the toy model the effective Lagrangian takes the general form
where the sum is over interactions
${\mathcal{O}}_{id}$
, involving
$i$
powers of the dimensionless field
$\theta $
and
$d$
derivatives. The power of
${m}_{R}$
premultiplying each term is chosen to ensure that the coefficient
${c}_{id}$
is dimensionless. (For instance, the interaction
$({\partial}_{\mu}\theta {\partial}^{\mu}\theta {)}^{2}$
has
$i=d=4$
.) There are three useful properties which all of the operators in this sum must satisfy:

1.
$d$
must be even by virtue of Lorentz invariance.

2.
Since the sum is only over interactions, it does not include the kinetic term, which is the unique term for which
$d=i=2$
.

3.
The
$U\left(1\right)$
symmetry implies every factor of
$\theta $
is differentiated at least once, and so
$d\ge i$
.
Furthermore, any term linear in
$\theta $
must therefore be a total derivative, and so may be omitted, implying
$i\ge 2$
without loss.
2.5.1 Powercounting lowenergy Feynman graphs
It is straightforward to track the powers of
$v$
and
${m}_{R}$
that interactions of the form ( 15 ) contribute to an
$\ell $
loop contribution to the amplitude
${\mathcal{A}}_{\mathcal{\mathcal{E}}}\left(E\right)$
for the scattering of
${N}_{i}$
initial Goldstone bosons into
${N}_{f}$
final Goldstone bosons at centreofmass energy
$E$
. The label
$\mathcal{\mathcal{E}}={N}_{i}+{N}_{f}$
here denotes the number of external lines in the corresponding graph. (The steps presented in this section closely follow the discussion of [
28]
.) With the desire of also being able to include later examples, consider the following slight generalization of the Lagrangian of Equation ( 15 ):
$$\begin{array}{c}{\mathcal{\mathcal{L}}}_{eff}={f}^{4}{\sum}_{n}\frac{{c}_{n}}{{M}^{{d}_{n}}}{\mathcal{O}}_{n}\left(\frac{\phi}{v}\right).\end{array}$$ 
(16)

Here
$\phi $
denotes a generic boson field,
${c}_{n}$
are again dimensionless coupling constants which we imagine to be at most
$\mathcal{O}\left(1\right)$
, and
$f$
,
$M$
, and
$v$
are mass scales of the underlying problem. The sum is again over operators which are powers of the fields and their derivatives, and
${d}_{n}$
is the dimension of the operator
${\mathcal{O}}_{n}$
, in powers of mass. For example, in the toymodel application we have
${f}^{2}=v{m}_{R}$
,
$M={m}_{\mathcal{\mathcal{R}}}$
, and we have written
$\theta =\phi /v$
. In the toymodel example the sum over
$n$
corresponds to the sum over
$i$
and
$d$
, and
${d}_{n}=d$
.
Imagine using this Lagrangian to compute a scattering amplitude
${\mathcal{A}}_{\mathcal{\mathcal{E}}}\left(E\right)$
involving the scattering of
$\mathcal{\mathcal{E}}$
relativistic particles whose energy and momenta are of order
$E$
. We wish to focus on the contribution to
$\mathcal{A}$
due to a Feynman graph having
$I$
internal lines and
${V}_{ik}$
vertices. The labels
$i$
and
$k$
here indicate two characteristics of the vertices:
$i$
counts the number of lines which converge at the vertex, and
$k$
counts the power of momentum which appears in the vertex. Equivalently,
$i$
counts the number of powers of the fields
$\phi $
which appear in the corresponding interaction term in the Lagrangian, and
$k$
counts the number of derivatives of these fields which appear there.
Some useful identities
The positive integers
$I$
,
$E$
, and
${V}_{ik}$
, which characterize the Feynman graph in question are not all independent since they are related by the rules for constructing graphs from lines and vertices.
The first such relation can be obtained by equating two equivalent ways of counting how internal and external lines can end in a graph. On the one hand, since all lines end at a vertex, the number of ends is given by summing over all of the ends which appear in all of the vertices:
${\sum}_{ik}i{V}_{ik}$
. On the other hand, there are two ends for each internal line, and one end for each external line in the graph:
$2I+E$
. Equating these gives the identity which expresses the `conservation of ends':
$$\begin{array}{c}2I+E={\sum}_{ik}i{V}_{ik},\text{conservation of ends}.\end{array}$$ 
(17)

A second useful identity gives the number of loops
$L$
for each (connected) graph:
$$\begin{array}{c}L=1+I{\sum}_{ik}{V}_{ik},\text{definition of}L.\end{array}$$ 
(18)

For simple planar graphs, this last equation agrees with the intuitive notion of what the number of loops in a graph means, since it expresses a topological invariant which states how the Euler number for a disc can be expressed in terms of the number of edges, corners, and faces of the triangles in one of its triangularization. For graphs which do not have the topology of a plane, Equation ( 18 ) should instead be read as defining the number of loops.
Dimensional estimates
We now collect the dependence of
${\mathcal{A}}_{\mathcal{\mathcal{E}}}\left(a\right)$
on the parameters appearing in
${\mathcal{\mathcal{L}}}_{eff}$
. Reading the Feynman rules from the Lagrangian of Equation ( 16 ) shows that the vertices in the Feynman graph contribute the following factor:
$$\begin{array}{c}(Vertex)={\prod}_{ik}{\left[i\left(2\pi {)}^{4}{\delta}^{4}\right(p){\left(\frac{p}{M}\right)}^{k}\left(\frac{{f}^{4}}{{v}^{i}}\right)\right]}^{{V}_{ik}},\end{array}$$ 
(19)

where
$p$
generically denotes the various momenta running through the vertex. Similarly, there are
$I$
internal lines in the graph, each of which contributes the additional factor:
$$\begin{array}{c}(Internalline)=\left[i\int \frac{{d}^{4}p}{(2\pi {)}^{4}}\left(\frac{{M}^{2}{v}^{2}}{{f}^{4}}\right)\frac{1}{{p}^{2}+{m}^{2}}\right],\end{array}$$ 
(20)

where, again,
$p$
denotes the generic momentum flowing through the line.
$m$
generically denotes the mass of any light particles which appear in the effective theory, and it is assumed that the kinetic terms which define their propagation are those terms in
${\mathcal{\mathcal{L}}}_{eff}$
involving two derivatives and two powers of the fields
$\phi $
.
As usual for a connected graph, all but one of the momentumconserving
$delta$
functions in Equation ( 19 ) can be used to perform one of the momentum integrals in Equation ( 20 ). The one remaining
$delta$
function which is left after doing so depends only on the external momenta
${\delta}^{4}\left(q\right)$
, and expresses the overall conservation of fourmomentum for the process. Future formulae are less cluttered if this factor is extracted once and for all, by defining the reduced amplitude
$\stackrel{~}{\mathcal{A}}$
by
$$\begin{array}{c}{\mathcal{A}}_{\mathcal{\mathcal{E}}}\left(E\right)=i\left(2\pi {)}^{4}{\delta}^{4}\right(q\left){\stackrel{~}{\mathcal{A}}}_{\mathcal{\mathcal{E}}}\right(E).\end{array}$$ 
(21)

Here
$q$
generically represents the external fourmomenta of the process, whose components are of order
$E$
in size.
The number of fourmomentum integrations which are left after having used all of the momentumconserving
$delta$
functions is then
$I{\sum}_{ik}{V}_{ik}+1=L$
. This last equality uses the definition, Equation ( 18 ), of the number of loops
$L$
.
We now estimate the result of performing the integration over the internal momenta. To do so it is most convenient to regulate the ultraviolet divergences which arise using dimensional regularization
^{
$\text{2}$
}
. For dimensionallyregularized integrals, the key observation is that the size of the result is set on dimensional grounds by the light masses or external momenta of the theory. That is, if all external energies
$q$
are comparable to (or larger than) the masses
$m$
of the light particles whose scattering is being calculated, then
$q$
is the light scale controlling the size of the momentum integrations, so dimensional analysis implies that an estimate of the size of the momentum integrations is
$$\begin{array}{c}\int \cdots \int {\left(\frac{{d}^{n}p}{(2\pi {)}^{n}}\right)}^{A}\frac{{p}^{B}}{({p}^{2}+{q}^{2}{)}^{C}}\sim {\left(\frac{1}{4\pi}\right)}^{2A}{q}^{nA+B2C},\end{array}$$ 
(22)

with a dimensionless prefactor which depends on the dimension
$n$
of spacetime, and which may be singular in the limit that
$n\to 4$
. Notice that the assumption that
$q$
is the largest relevant scale in the lowenergy theory explicitly excludes the case of the scattering of nonrelativistic particles.
One might worry whether such a simple dimensional argument can really capture the asymptotic dependence of a complicated multidimensional integral whose integrand is rife with potential singularities. The ultimate justification for this estimate lies with general results like Weinberg's theorem [
142,
84,
128,
78]
, which underly the powercounting analyses of renormalizability. These theorems ensure that the simple dimensional estimates capture the correct behaviour up to logarithms of the ratios of highenergy and lowenergy mass scales.
With this estimate for the size of the momentum integrations, we find the following contribution to the amplitude
${\stackrel{~}{\mathcal{A}}}_{\mathcal{\mathcal{E}}}\left(E\right)$
:
$$\begin{array}{c}\int \cdots \int {\left(\frac{{d}^{4}p}{(2\pi {)}^{4}}\right)}^{L}\frac{{p}^{X}}{({p}^{2}+{q}^{2}{)}^{I}}\sim {\left(\frac{1}{4\pi}\right)}^{2L}{q}^{Y},\end{array}$$ 
(23)

where
$X={\sum}_{ik}k{V}_{ik}$
and
$Y=4L2I+{\sum}_{ik}k{V}_{ik}$
. Liberal use of the identities ( 17 ) and ( 18 ), and taking
$q\sim E$
, allows this to be rewritten as the following estimate:
$$\begin{array}{c}{\stackrel{~}{\mathcal{A}}}_{\mathcal{\mathcal{E}}}\left(E\right)\sim {f}^{4}{\left(\frac{1}{v}\right)}^{\mathcal{\mathcal{E}}}{\left(\frac{{M}^{2}}{4\pi {f}^{2}}\right)}^{2L}{\left(\frac{E}{M}\right)}^{P},\end{array}$$ 
(24)

with
$P=2+2L+{\sum}_{ik}(k2){V}_{ik}$
. Equivalently, if we group terms depending on
$L$
, Equation ( 24 ) may also be written as
$$\begin{array}{c}{\stackrel{~}{\mathcal{A}}}_{\mathcal{\mathcal{E}}}\left(E\right)\sim {f}^{4}{\left(\frac{1}{v}\right)}^{\mathcal{\mathcal{E}}}{\left(\frac{ME}{4\pi {f}^{2}}\right)}^{2L}{\left(\frac{E}{M}\right)}^{{P}^{\prime}},\end{array}$$ 
(25)

with
${P}^{\prime}=2+{\sum}_{ik}(k2){V}_{ik}$
.
Equation (
24 ) is the principal result of this section. Its utility lies in the fact that it links the contributions of the various effective interactions in the effective Lagrangian ( 16 ) with the dependence of observables on small energy ratios such as
$r=E/M$
. As a result it permits the determination of which interactions in the effective Lagrangian are required to reproduce any given order in
$E/M$
in physical observables.
Most importantly, Equation (
24 ) shows how to calculate using nonrenormalizable theories. It implies that even though the Lagrangian can contain arbitrarily many terms, and so potentially arbitrarily many coupling constants, it is nonetheless predictive so long as its predictions are only made for lowenergy processes, for which
$E/M\ll 1$
. (Notice also that the factor
$(M/f{)}^{4L}$
in Equation ( 24 ) implies, all other things being equal, that the scale
$f$
cannot be taken to be systematically smaller than
$M$
without ruining the validity of the loop expansion in the effective lowenergy theory.)
2.5.2 Application to the toy model
We now apply this powercounting estimate to the toy model discussed earlier. Using the relations
${f}^{2}=v{m}_{\mathcal{\mathcal{R}}}$
and
$M={m}_{\mathcal{\mathcal{R}}}$
, we have
$$\begin{array}{c}{\mathcal{A}}_{\mathcal{\mathcal{E}}}\left(E\right)\sim {v}^{2}{m}_{\mathcal{\mathcal{R}}}^{2}{\left(\frac{1}{v}\right)}^{\mathcal{\mathcal{E}}}{\left(\frac{{m}_{\mathcal{\mathcal{R}}}}{4\pi v}\right)}^{2L}{\left(\frac{E}{{m}_{\mathcal{\mathcal{R}}}}\right)}^{P},\end{array}$$ 
(26)

where
$P=2+2L+{\sum}_{id}(d2){V}_{id}$
. As above,
${V}_{id}$
counts the number of times an interaction involving
$i$
powers of fields and
$d$
derivatives appears in the amplitude. An equivalent form for this expression is
$$\begin{array}{c}{\mathcal{A}}_{\mathcal{\mathcal{E}}}\left(E\right)\sim {v}^{2}{E}^{2}{\left(\frac{1}{v}\right)}^{\mathcal{\mathcal{E}}}{\left(\frac{E}{4\pi v}\right)}^{2L}{\prod}_{i}{\prod}_{d>2}{\left(\frac{E}{{m}_{\mathcal{\mathcal{R}}}}\right)}^{(d2){V}_{id}}.\end{array}$$ 
(27)

Equations ( 26 ) and ( 27 ) have several noteworthy features:

∙
The condition
$d\ge i\ge 2$
ensures that all of the powers of
$E$
appearing in
${\mathcal{A}}_{\mathcal{\mathcal{E}}}$
are positive.
Furthermore, since
$d=2$
only occurs for the kinetic term,
$d\ge 4$
for all interactions, implying
${\mathcal{A}}_{\mathcal{\mathcal{E}}}$
is suppressed by at least 4 powers of
$E$
, and higher loops cost additional powers of
$E$
.
Furthermore, it is straightforward to identify the graphs which contribute to
${\mathcal{A}}_{\mathcal{\mathcal{E}}}$
to any fixed power of
$E$
.

∙
As is most clear from Equation ( 27 ), perturbation theory in the effective theory relies only on the assumptions
$E\ll 4\pi v$
and
$E\ll {m}_{\mathcal{\mathcal{R}}}$
. In particular, it does not rely on the ratio
${m}_{\mathcal{\mathcal{R}}}/\left(4\pi v\right)$
being small. Since
${m}_{\mathcal{\mathcal{R}}}=\lambda v$
in the underlying toy model, this ratio is simply of order
$\lambda /\left(4\pi \right)$
, showing how weak coupling for the Goldstone boson is completely independent of the strength of the couplings in the underlying theory.
To see how Equations ( 26 ) and ( 27 ) are used, consider the first few powers of
$E$
in the toy model. For any
$\mathcal{\mathcal{E}}$
the leading contributions for small
$E$
come from tree graphs,
$L=0$
. The tree graphs that dominate are those for which
${\sum}_{id}^{\prime}(d2){V}_{id}$
takes the smallest possible value. For example, for 2particle scattering
$\mathcal{\mathcal{E}}=4$
, and so precisely one tree graph is possible for which
${\sum}_{id}^{\prime}(d2){V}_{id}=2$
, corresponding to
${V}_{44}=1$
and all other
${V}_{id}=0$
. This identifies the single graph which dominates the 4point function at low energies, and shows that the resulting leading energy dependence is
${\mathcal{A}}_{4}\left(E\right)\sim {E}^{4}/\left({v}^{2}{m}_{R}^{2}\right)$
.
The utility of powercounting really becomes clear when subleading behaviour is computed, so consider the size of the leading corrections to the 4point scattering amplitude. Order
${E}^{6}$
contributions are achieved if and only if either (i)
$L=1$
and
${V}_{i4}=1$
with all others zero, or (ii)
$L=0$
and
${\sum}_{i}\left(4{V}_{i6}+2{V}_{i4}\right)=4$
. Since there are no
$d=2$
interactions, no oneloop graphs having 4 external lines can be built using precisely one
$d=4$
vertex, and so only tree graphs can contribute. Of these, the only two choices allowed by
$\mathcal{\mathcal{E}}=4$
at order
${E}^{6}$
are therefore the choices
${V}_{46}=1$
or
${V}_{34}=2$
. Both of these contribute a result of order
${\mathcal{A}}_{4}\left(E\right)\sim {E}^{6}/\left({v}^{2}{m}_{\mathcal{\mathcal{R}}}^{4}\right)$
.
2.6 The effective Lagrangian logic
With the powercounting results in hand we can see how to calculate predictively – including loops – using the nonrenormalizable effective theory. The logic follows these steps:

1.
Choose the accuracy desired in the answer. (For instance an accuracy of 1% might be desired in a particular scattering amplitude.)

2.
Determine the order in the small ratio of scales (i.e.,
$r=E/{m}_{\mathcal{\mathcal{R}}}$
in the toy model) which is required in order to achieve the desired accuracy. (For instance if
$r=0.1$
then
$\mathcal{O}\left({r}^{2}\right)$
is required to achieve 1% accuracy.)

3.
Use the powercounting results to identify which terms in
${\mathcal{\mathcal{L}}}_{eff}$
can contribute to the observable of interest to the desired order in
$r$
. At any fixed order in
$r$
this always requires a finite number (say:
$N$
) of terms in
${\mathcal{\mathcal{L}}}_{eff}$
which can contribute.

4.

(a)
If the underlying theory is known, and is calculable, then compute the required coefficients of the
$N$
required effective interactions to the accuracy required. (In the toy model this corresponds to calculating the coefficients
$a$
,
$b$
,
$c$
, etc.)

(b)
If the underlying theory is unknown, or is too complicated to permit the calculation of
${\mathcal{\mathcal{L}}}_{eff}$
, then leave the
$N$
required coefficients as free parameters. The procedure is nevertheless predictive if more than
$N$
observables can be identified whose predictions depend only on these parameters.
Step 4a is required when the lowenergy expansion is being used as an efficient means to accurately calculating observables in a wellunderstood theory. It is the option of choosing instead Step 4b , however, which introduces much of the versatility of effectiveLagrangian methods. Step 4b is useful both when the underlying theory is not known (such as when searching for physics beyond the Standard Model) and when the underlying physics is known but complicated (like when describing the lowenergy interactions of pions in quantum chromodynamics).
The effective Lagrangian is in this way seen to be predictive even though it is not renormalizable in the usual sense. In fact, renormalizable theories are simply the special case of Step
4b where one stops at order
${r}^{0}$
, and so are the ones which dominate in the limit that the light and heavy scales are very widely separated. We see in this way why renormalizable interactions play ubiquitous roles through physics! These observations have important conceptual implications for the quantum behaviour of other nonrenormalizable theories, such as gravity, to which we return in the next Section 3 .
2.6.1 The choice of variables
The effective Lagrangian of the toy model seems to carry much more information when
$\theta $
is used to represent the light particles than it would if
$\mathcal{\mathcal{I}}$
were used. How can physics depend on the fields which are used to parameterize the theory?
Physical quantities do not depend on what variables are used to describe them, and the lowenergy scattering amplitude is suppressed by the same power of
$r$
in the toy model regardless of whether it is the effective Lagrangian for
$\mathcal{\mathcal{I}}$
or
$\theta $
which is used at an intermediate stage of the calculation.
The final result would nevertheless appear quite mysterious if
$\mathcal{\mathcal{I}}$
were used as the lowenergy variable, since it would emerge as a cancellation only at the end of the calculation. With
$\theta $
the result is instead manifest at every step. Although the physics does not depend on the variables in terms of which it is expressed, it nevertheless pays mortal physicists to use those variables which make manifest the symmetries of the underlying system.
2.6.2 Regularization dependence
The definition of
${\mathcal{\mathcal{L}}}_{eff}$
appears to depend on lots of calculational details, like the value of
$\Lambda $
(or, in dimensional regularization, the matching scale) and the minutae of how the cutoff is implemented.
Why doesn't
${\mathcal{\mathcal{L}}}_{eff}$
depend on all of these details?
${\mathcal{\mathcal{L}}}_{eff}$
generally does depend on all of the regularizational details. But these details all must cancel in final expressions for physical quantities. Thus, some
$\Lambda $
dependence enters into scattering amplitudes through the explicit dependence which is carried by the couplings of
${\mathcal{\mathcal{L}}}_{eff}$
(beyond tree level). But
$\Lambda $
also potentially enters scattering amplitudes because loops over all light degrees of freedom must be cut off at
$\Lambda $
in the effective theory, by definition. The cancellation of these two sources of cutoffdependence is guaranteed by the observation that
$\Lambda $
enters only as a bookmark, keeping track of the light and heavy degrees of freedom at intermediate steps of the calculation.
This cancellation of
$\Lambda $
in all physical quantities ensures that we are free to make any choice of cutoff which makes the calculation convenient. After all, although all regularization schemes for
${\mathcal{\mathcal{L}}}_{eff}$
give the same answers, more work is required for some schemes than for others. Again, mere mortal physicists use an inconvenient scheme at their own peril!
Dimensional regularization
This freedom to use any convenient scheme is ultimately the reason why dimensional regularization may be used when defining lowenergy effective theories, even though the dimensionallyregularized effective theories involve fields with modes of arbitrarily high momentum. At first sight the necessity of having modes of arbitrarily large momenta appear in dimensionallyregularized theories would seem to make dimensional regularization inconsistent with effective field theory. After all, any dimensionallyregularized lowenergy theory would appear necessarily to include states having arbitrarily high energies.
In practice this is not a problem, so long as the effective interactions are chosen to properly reproduce the dimensionallyregularized scattering amplitudes of the full theory (orderbyorder in
$1/M$
). This is possible ultimately because the difference between the cutoffand dimensionallyregularized lowenergy theory can itself be parameterized by appropriate local effective couplings within the lowenergy theory. Consequently, any regularizationdependent properties will necessarily drop out of final physical results, once the (renormalized) effective couplings are traded for physical observables.
In practice this means that one does not construct a dimensionallyregularized effective theory by explicitly performing a path integral over successively higherenergy momentum modes of all fields in the underlying theory. Instead one defines effective dimensionally regularized theories for which heavy fields are completely removed. For instance, suppose it is the lowenergy influence of a heavy particle
$h$
having mass
$M$
which is of interest. Then the highenergy theory consists of a dimensionallyregularized collection of light fields
${\ell}_{i}$
and
$h$
, while the effective theory is a dimensionallyregularized theory of the light fields
${\ell}_{i}$
only. The effective couplings of the lowenergy theory are obtained by performing a matching calculation, whereby the couplings of the lowenergy effective theory are chosen to reproduce scattering amplitudes or Green's functions of the underlying theory orderbyorder in powers of the inverse heavy scale
$1/M$
. Once the couplings of the effective theory are determined in this way in terms of those of the underlying fundamental theory, they may be used to compute any purely lowenergy observable.
An important technical point arises if calculations are being done to oneloop accuracy (or more) using dimensional regularization. For these calculations it is convenient to trade the usual minimalsubtraction (or modifiedminimalsubtraction) renormalization scheme, for a slightly modified
decoupling subtraction scheme [
148,
123,
124]
. In this scheme couplings are defined using minimal (or modifiedminimal) subtraction between successive particle threshholds, with the couplings matched from the underlying theory to the effective theory as each heavy particle is successively integrated out. This results in a renormalization group evolution of effective couplings which is almost as simple as for minimal subtraction, but with the advantage that the implications of heavy particles in running couplings are explicitly decoupled as one passes to energies below the heavy particle mass.
Some textbooks which describe effective Lagrangians are [
73,
58]
; some reviews articles which treat lowenergy effective field theories (mostly focussing on pion interactions) are [
116,
107,
113,
132,
126,
99,
74]
.
A great advantage of the dimensionallyregularized effective theory is the absence of the cutoff scale
$\Lambda $
, which implies that the only dimensionful scales which arise are physical particle masses.
This was implicitly used in the powercounting arguments given earlier, wherein integrals over loop momenta were replaced by powers of heavy masses on dimensional grounds. This gives a sufficiently accurate estimate despite the ultraviolet divergences in these integrals, provided the integrals are dimensionally regularized. For effective theories it is powers of the arbitrary cutoff scale
$\Lambda $
which would arise in these estimates, and because
$\Lambda $
cancels out of physical quantities, this just obscures how heavy physical masses appear in the final results.
2.7 The meaning of renormalizability
The previous discussion about the cancellation between the cutoffs on virtual lightparticle momenta and the explicit cutoffdependence of
${\mathcal{\mathcal{L}}}_{eff}$
is eerily familiar. It echoes the traditional discussion of the cancellation of the regularized ultraviolet divergences of loop integrals against the regularization dependence of the counterterms of the renormalized Lagrangian. There are, however, the following important differences:

∙
The cancellations in the effective theory occur even though
$\Lambda $
is not sent to infinity, and even though
${\mathcal{\mathcal{L}}}_{eff}$
contains arbitrarily many terms which are not renormalizable in the traditional sense (i.e., terms whose coupling constants have dimensions of inverse powers of mass in fundamental units where
$\u0127=c=1$
).

∙
Whereas the cancellation of regularization dependence in the traditional renormalization picture appears adhoc and implausible, those in the effective Lagrangian are sweet reason personified. This is because they simply express the obvious fact that
$\Lambda $
only was introduced as an intermediate step in a calculation, and so cannot survive uncancelled in the answer.
This resemblance suggests Wilson's physical reinterpretation of the renormalization procedure.
Rather than considering a model's classical Lagrangian, such as
$\mathcal{\mathcal{L}}$
of Equation ( 1 ), as something pristine and fundamental, it is better to think of it also as an effective Lagrangian obtained by integrating out still more microscopic degrees of freedom. The cancellation of the ultraviolet divergences in this interpretation is simply the usual removal of an intermediate step in a calculation to whose microscopic part we are not privy.
3 LowEnergy Quantum Gravity
According to the approach just described, nonrenormalizable theories are not fundamentally different from renormalizable ones. They simply differ in their sensitivity to more microscopic scales which have been integrated out. It is instructive to see what this implies for the nonrenormalizable theories which sometimes are required to successfully describe experiments. This is particularly true for the most famous such case, Einstein's theory of gravity. (See [
57]
for another pedagogical review of gravity as an effective theory.)
3.1 General relativity as an effective theory
The lowenergy degrees of freedom in this case are the metric
${g}_{\mu \nu}$
of spacetime itself. As has been seen in previous sections, Einstein's action for this theory should be considered to be just one term in a sum of all possible interactions which are consistent with the symmetries of the lowenergy theory (which in this case are: general covariance and local Lorentz invariance). Organizing the resulting action into powers of derivatives of the metric leads to the following effective Lagrangian:
$$\begin{array}{c}\frac{{\mathcal{\mathcal{L}}}_{eff}}{\sqrt{g}}=\lambda +\frac{{M}_{p}^{2}}{2}R+a{R}_{\mu \nu}{R}^{\mu \nu}+b{R}^{2}+d{R}_{\mu \nu \lambda \rho}{R}^{\mu \nu \lambda \rho}+e\square R+\frac{c}{{m}^{2}}{R}^{3}+....\end{array}$$ 
(28)

Here
${R}_{\mu \nu \lambda \rho}$
is the metric's Riemann tensor,
${R}_{\mu \nu}$
is its Ricci tensor, and
$R$
is the Ricci scalar, each of which involves precisely two derivatives of the metric. For brevity only one representative example of the many possible curvaturecubed terms is explicitly written. (We use here Weinberg's curvature conventions [
146]
, which differ from those of Misner, Thorne, and Wheeler [
121]
by an overall sign.) The first term in Equation ( 28 ) is the cosmological constant, which is dropped in what follows since the observed size of the universe implies
$\lambda $
is extremely small. There is, of course, no real theoretical understanding why the cosmological constant should be this small (a comprehensive review of the cosmological constant problem is given in [
151]
; for a recent suggestion in the spirit of effective field theories see [
3,
29]
). Once the cosmological term is dropped, the leading term in the derivative expansion is the one linear in
$R$
, which is the usual Einstein–Hilbert action of general relativity. Its coefficient defines Newton's constant (and so also the Planck mass,
${M}_{p}^{2}=8\pi G$
).
The explicit mass scales,
$m$
and
${M}_{p}$
, are introduced to ensure that the remaining constants
$a$
,
$b$
,
$c$
,
$d$
and
$e$
appearing in Equation ( 28 ) are dimensionless. Since it appears in the denominator, the mass scale
$m$
can be considered as the smallest microscopic scale to have been integrated out to obtain Equation ( 28 ). For definiteness we might take the electron mass,
$m=5\times {10}^{4}GeV$
, for
$m$
when considering applications at energies below the masses of all elementary particles. (Notice that contributions like
${m}^{2}R$
or
${R}^{3}/{M}_{p}^{2}$
could also exist, but these are completely negligible compared to the terms displayed in Equation ( 28 ).)
3.1.1 Redundant interactions
As discussed in the previous section, some of the interactions in the Lagrangian ( 28 ) may be redundant, in the sense that they do not contribute independently to physical observables (like graviton scattering amplitudes about some fixed geometry, say). To eliminate these we are free to drop any terms which are either total derivatives or which vanish when evaluated at solutions to the lowerorder equations of motion.
The freedom to drop total derivatives allows us to set the couplings
$d$
and
$e$
to zero. We can drop
$e$
because
$\sqrt{g}\square R={\partial}_{\mu}\left[\sqrt{g}{\nabla}^{\mu}R\right]$
, and we can drop
$d$
because the quantity
$$\begin{array}{c}X={R}_{\mu \nu \lambda \rho}{R}^{\mu \nu \lambda \rho}4{R}_{\mu \nu}{R}^{\mu \nu}+{R}^{2},\end{array}$$ 
(29)

integrates to give a topological invariant in 4 dimensions. That is, in 4 dimensions
$\chi \left(M\right)=(1/32{\pi}^{2}){\int}_{M}\sqrt{g}X{d}^{4}x$
gives the Euler number of a compact manifold
$M$
– and so
$X$
is locally a total derivative. It is therefore always possible to replace, for example,
${R}_{\mu \nu \lambda \rho}{R}^{\mu \nu \lambda \rho}$
in the effective Lagrangian with the linear combination
$4{R}_{\mu \nu}{R}^{\mu \nu}{R}^{2}$
, with no consequences for any observables which are insensitive to the overall topology of spacetime (such as the classical equations, or perturbative particle interactions). Any such observable therefore is unchanged under the replacement
$(a,b,d)\to ({a}^{\prime},{b}^{\prime},{d}^{\prime})=(ad,b+4d,0)$
.
The freedom to perform field redefinitions allows further simplification (just as was found for the toy model in earlier sections). To see how this works, consider the infinitesimal field redefinition
$\delta {g}_{\mu \nu}={Y}_{\mu \nu}$
, under which the leading term in
${\mathcal{\mathcal{L}}}_{eff}$
undergoes the variation
$$\begin{array}{c}\frac{{M}_{p}^{2}}{2}\delta \int {d}^{4}x\sqrt{g}R=\frac{{M}_{p}^{2}}{2}\int {d}^{4}x\sqrt{g}\left[{R}^{\mu \nu}\frac{1}{2}R{g}^{\mu \nu}\right]{Y}_{\mu \nu}.\end{array}$$ 
(30)

In particular, we may set the constants
$a$
and
$b$
to zero simply by choosing
${M}_{p}^{2}{Y}_{\mu \nu}=2a{R}_{\mu \nu}(a+2b)R{g}_{\mu \nu}$
. Since the variation of the lowerorder terms in the action are always proportional to their equations of motion, quite generally any term in
${\mathcal{\mathcal{L}}}_{eff}$
which vanishes on use of the lowerorder equations of motion can be removed in this way (order by order in
$1/m$
and
$1/{M}_{p}$
).
Since the lowestorder equations of motion for pure gravity (without a cosmological constant) imply
${R}_{\mu \nu}=0$
, we see that all of the interactions beyond the Einstein–Hilbert term which are explicitly written in Equation ( 28 ) can be removed in one of these two ways. The first interaction which can have physical effects (for pure gravity with no cosmological constant) in this lowenergy expansion is therefore proportional to the cube of the Riemann tensor.
This last conclusion changes if matter or a cosmological constant are present, however, since then the lowestorder field equations become
${R}_{\mu \nu}={S}_{\mu \nu}$
for some nonzero tensor
${S}_{\mu \nu}$
. Then terms like
${R}^{2}$
or
${R}_{\mu \nu}{R}^{\mu \nu}$
no longer vanish when evaluated at the solutions to the equations of motion, but are instead equivalent to interactions of the form
$({S}_{\mu}^{\mu}{)}^{2}$
,
${S}_{\mu \nu}{R}^{\mu \nu}$
, or
${S}_{\mu \nu}{S}^{\mu \nu}$
. Since some of our later applications of
${\mathcal{\mathcal{L}}}_{eff}$
are to the gravitational potential energy of various localized energy sources, we shall find that these terms can generate contact interactions amongst these sources.
3.2 Power counting
Since gravitons are weakly coupled, perturbative powercounting may be used to see how the highenergy scales
${M}_{p}$
and
$m$
enter into observables like graviton scattering amplitudes about some fixed macroscopic metric. We now perform this power counting along the lines of previous sections for the interactions of gravitons near flat space:
${g}_{\mu \nu}={\eta}_{\mu \nu}+{h}_{\mu \nu}$
. For the purposes of this power counting all we need to know about the curvatures is that they each involve all possible powers of – but only two derivatives of – the fluctuation field
${h}_{\mu \nu}$
.
A powercounting estimate for the
$L$
loop contribution to the
$\mathcal{\mathcal{E}}$
point gravitonscattering amplitude
${\mathcal{A}}_{\mathcal{\mathcal{E}}}$
, which involves
${V}_{id}$
vertices involving
$d$
derivatives and the emission or absorption of
$i$
gravitons, may be found by arguments identical to those used previously for the toy model. The main difference from the toymodel analysis is the existence for gravity of interactions involving two derivatives, which all come from the Einstein–Hilbert term in
${\mathcal{\mathcal{L}}}_{eff}$
. (Such terms also arise for Goldstone bosons for symmetrybreaking patterns involving nonAbelian groups and are easily incorporated into the analysis.) The resulting estimate for
${\mathcal{A}}_{\mathcal{\mathcal{E}}}$
turns out to be of order
$$\begin{array}{c}{\mathcal{A}}_{\mathcal{\mathcal{E}}}\left(E\right)\sim {m}^{2}{M}_{p}^{2}{\left(\frac{1}{{M}_{p}}\right)}^{\mathcal{\mathcal{E}}}{\left(\frac{m}{4\pi {M}_{p}}\right)}^{2L}{\left(\frac{{m}^{2}}{{M}_{p}^{2}}\right)}^{Z}{\left(\frac{E}{m}\right)}^{P},\end{array}$$ 
(31)

where
$Z={\sum}_{id}^{\prime}{V}_{id}$
and
$P=2+2L+{\sum}_{id}^{\prime}(d2){V}_{id}$
. The prime on both of these sums indicates the omission of the case
$d=2$
from the sum over
$d$
. If we instead group the terms involving powers of
$L$
and
${V}_{ik}$
, Equation ( 31 ) takes the equivalent form
$$\begin{array}{c}{\mathcal{A}}_{\mathcal{\mathcal{E}}}\left(E\right)\sim {E}^{2}{M}_{p}^{2}{\left(\frac{1}{{M}_{p}}\right)}^{\mathcal{\mathcal{E}}}{\left(\frac{E}{4\pi {M}_{p}}\right)}^{2L}{\prod}_{i}{\prod}_{d>2}{\left[\frac{{E}^{2}}{{M}_{p}^{2}}{\left(\frac{E}{m}\right)}^{d4}\right]}^{{V}_{id}}.\end{array}$$ 
(32)

Notice that since
$d$
is even, the condition
$d>2$
in the product implies there are no negative powers of
$E$
in this expression.
Equations (
31 ) and ( 32 ) share many of the noteworthy features of Equations ( 26 ) and ( 27 ).
Again the weakness of the graviton's coupling follows only from the lowenergy approximations,
$E\ll {M}_{p}$
and
$E\ll m$
. When written as in Equation ( 32 ), it is clear that even though the ratio
$E/m$
is potentially much larger than
$E/{M}_{p}$
, it does not actually arise in
${\mathcal{A}}_{\mathcal{\mathcal{E}}}$
unless contributions from at least curvaturecubed interactions are included (for which
$d=6$
).
These expressions permit a determination of the dominant lowenergy contributions to scattering amplitudes. The minimum suppression comes when
$L=0$
and
$P=2$
, and so is given by arbitrary tree graphs constructed purely from the Einstein–Hilbert action. We are led in this way to what we are in any case inclined to believe: It is classical general relativity which governs the lowenergy dynamics of gravitational waves!
But the nexttoleading contributions are also quite interesting. These arise in one of two ways, either
(i)
$L=1$
and
${V}_{id}=0$
for any
$d\ne 2$
, or (ii)
$L=0$
,
${\sum}_{i}{V}_{i4}=1$
,
${V}_{i2}$
is arbitrary, and all other
${V}_{id}$
vanish. That is, the next to leading contribution is obtained by computing the oneloop corrections using only Einstein gravity, or by working to tree level and including precisely one curvaturesquared interaction in addition to any number of interactions from the Einstein–Hilbert term. Both are suppressed compared to the leading term by a factor of
$(E/{M}_{p}{)}^{2}$
, and the oneloop contribution carries an additional factor of
$(1/4\pi {)}^{2}$
.
For instance, for 2body graviton scattering we have
$\mathcal{\mathcal{E}}=4$
, and so the above arguments imply the leading behaviour is
${\mathcal{A}}_{4}\left(E\right)\sim {A}_{2}(E/{M}_{p}{)}^{2}+{A}_{4}(E/{M}_{p}{)}^{4}+...$
, where the numbers
${A}_{2}$
and
${A}_{4}$
have been explicitly calculated. At tree level all of the amplitudes turn out to vanish except for those which are related by crossing symmetry to the amplitude for which all graviton helicities have the same sign, and this is given by [
51]
:
$$\begin{array}{c}i{\mathcal{A}}_{(++,++)}^{tree}=8\pi G\left(\frac{{s}^{3}}{tu}\right),\end{array}$$ 
(33)

where
$s$
,
$t$
and
$u$
are the usual Mandelstam variables, all of which are proportional to the square of the centreofmass energy
${E}_{cm}$
. Besides vindicating the powercounting expectation that
$\mathcal{A}\sim (E/{M}_{p}{)}^{2}$
to leading order, this example also shows that the potentially framedependent energy
$E$
, which is relevant in the powercounting analysis, is in this case really the invariant centreofmass energy
${E}_{cm}$
.
The oneloop correction to this result has also been computed [
63]
, and is infrared divergent.
These infrared divergences cancel in the usual way with treelevel bremsstrahlung diagrams [
143]
, leading to a finite result [
61]
, which is suppressed as expected relative to the tree contribution by terms of order
$(E/4\pi {M}_{p}{)}^{2}$
, up to logarithmic corrections.
3.2.1 Including matter
The observables of most practical interest for experimental purposes involve the gravitational interactions of various kinds of matter. It is therefore useful to generalize the previous arguments to include matter and gravity coupled to one another. In most situations this generalization is reasonably straightforward, but somewhat paradoxically it is more difficult to treat the interactions of nonrelativistic matter than of relativistic matter. This section describes the reasons for this difference.
Relativistic matter
Consider first relativistic matter coupled to gravity. Rather than describing the general case, it suffices for the present purposes to consider instead a relativistic boson (such as a massless scalar or a photon) coupled to gravity but which does not selfinteract. The matter Lagrangian for such a system is then
$$\begin{array}{c}\frac{{\mathcal{\mathcal{L}}}_{mat}}{\sqrt{g}}=\frac{1}{2}{g}^{\mu \nu}{\partial}_{\mu}\phi {\partial}_{\nu}\phi +\frac{1}{4}{F}_{\mu \nu}{F}^{\mu \nu},\end{array}$$ 
(34)

and so the new interaction terms all involve at most two matter fields, two derivatives, and any number of undifferentiated metric fluctuations. This system is simple enough to include directly into the above analysis provided the gravitonmatter vertices are counted together with those from the Einstein–Hilbert term when counting the vertices having precisely two derivatives with
${V}_{i2}$
.
Particular kinds of higherderivative terms involving the matter fields may also be included equally trivially, provided the mass scales which appear in these terms appear in the same way as they did for the graviton. For instance, scalar functions built from arbitrary powers of
${A}_{\mu}/{M}_{p}$
and its derivatives
${\partial}_{\mu}/m$
can be included, along the lines of
$$\begin{array}{c}\Delta {\mathcal{\mathcal{L}}}_{n}={c}_{kn}{m}^{2}{M}_{p}^{2}{\left(\frac{{F}_{\mu \nu}{\square}^{k}{F}^{\mu \nu}}{{m}^{2+2k}{M}_{p}^{2}}\right)}^{n}.\end{array}$$ 
(35)

If the dimensionless constant
${c}_{kn}$
in these expressions is
$\mathcal{O}\left(1\right)$
, then the powercounting result for the dependence of amplitudes on
$m$
and
${M}_{p}$
is the same as it is for the puregravity theory, with vertices formed from
$\Delta {\mathcal{\mathcal{L}}}_{n}$
counted amongst those with
$d=(2+2k)n$
derivatives. If
$m\ll {M}_{p}$
it is more likely that powers of
${A}_{\mu}$
come suppressed by inverse powers of
$m$
rather than
${M}_{p}$
, however, in which case additional
${A}_{\mu}$
vertices are less suppressed than would be indicated above. The extension of the earlier powercounting estimate to this more general situation is straightforward.
Similar estimates also apply if a mass
${m}_{\phi}$
for the scalar field is included, provided that this mass is not larger than the energy flowing through the external lines:
${m}_{\phi}\lesssim E$
. This kind of mass does not change the powercounting result appreciably for observables which are infrared finite (which may require, as mentioned above summing over an indeterminate number of soft final gravitons).
They do not change the result because infraredfinite quantities are at most logarithmically singular as
${m}_{\phi}\to 0$
[
149]
, and so their expansion in
${m}_{\phi}/E$
simply adds terms for which factors of
$E$
are replaced by smaller factors of
${m}_{\phi}$
. But the above discussion can change dramatically if
${m}_{\phi}\gg E$
, since an important ingredient in the dimensional estimate is the assumption that the largest scale in the graph is the external energy
$E$
. Consequently the powercounting given above only directly applies to relativistic particles.
Nonrelativistic matter
The situation is more complicated if the matter particles move nonrelativistically, since in this case the particle mass is much larger than the momenta involved in the external lines,
$p=\mathbf{p}\ll {m}_{\phi}$
, so
$E\approx {m}_{\phi}+{p}^{2}/\left(2{m}_{\phi}\right)+...$
. We expect quantum corrections to the gravitational interactions of such particles also to be suppressed (such as, for instance, for the Earth) despite the energies and momenta involved being much larger than
${M}_{p}$
. Indeed, most of the tests of general relativity involve the gravitational interactions and orbits of very nonrelativistic `particles', like planets and stars. How can this be understood?
The case of nonrelativistic particles is also of real practical interest for the applications of effective field theories in other branches of physics. This is so, even though one might think that an effective theory should contain only particles which are very light. Nonrelativistic particles can nevertheless arise in practice within an effective field theory, even particles having masses which are large compared to those of the particles which were integrated out to produce the effective field theory in the first place. Such massive particles can appear consistently in a lowenergy theory provided they are stable (or extremely longlived), and so cannot decay and release enough energy to invalidate the lowenergy approximation. Some wellknown examples of this include the lowenergy nuclear interactions of nucleons (as described within chiral perturbation theory [
152,
153,
100,
109]
), the interactions of heavy fermions like the
$b$
and
$t$
quark (as described by heavyquark effective theory (HQET) [
90,
91,
108]
), and the interactions of electrons and nuclei in atomic physics (as described by nonrelativistic quantum electrodynamics (NRQED) [
34,
111,
103,
104,
127,
110]
).
The key to understanding the effective field theory for very massive, stable particles at low energies lies in the recognition that their antiparticles need
not be included since they would have already been integrated out to obtain the effective field theory of interest. As a result heavyparticle lines within the Feynman graphs of the effective theory only directly connect external lines, and never arise as closed loops.
The most direct approach to estimating the size of quantum corrections in this case is to powercount as before, subject to the restriction that graphs including internal closed loops of heavy particles are to be excluded. Donoghue and Torma [
60]
have performed such a powercounting analysis along these lines, and shows that quantum effects remain suppressed by powers of light particle energies (or small momentum transfers) divided by
${M}_{p}$
through the first few nontrivial orders of perturbation theory. Although heavyparticle momenta can be large,
$p\gg {M}_{p}$
, they only arise in physical quantities through the small relativistic parameter
$p/{m}_{\phi}\sim v$
rather than through
$p/{M}_{p}$
, extending the suppression of quantum effects obtained earlier to nonrelativistic problems.
Unfortunately, if a calculation is performed within a covariant gauge, individual Feynman graphs
can depend on large powers like
${m}_{\phi}/{M}_{p}$
, even though these all cancel in physical amplitudes.
For this reason an allorders inductive proof of the above powercounting remains elusive. As Donoghue and Torma [
60]
also make clear, progress towards such an allorders powercounting result is likely to be easiest within a physical, noncovariant gauge, since such a gauge allows powers of small quantities like
$v$
to be most easily followed.
Nonrelativistic effective field theory
If experience with electromagnetism is any guide, effective field theory techniques are also likely to be the most efficient way to systematically keep track of both the expansion in inverse powers of both heavy masses,
$1/{M}_{p}$
and
$1/{m}_{\phi}$
– particularly for bound orbits. Relative to the theories considered to this point, the effective field theory of interest has two unusual properties. First, since it involves very slowlymoving particles, Lorentz invariance is not simply realized on the corresponding heavyparticle fields. Second, since the effective theory does not contain antiparticles for the heavy particles, the heavy fields which describe them contain only positivefrequency parts.
To illustrate how these features arise, we briefly sketch how such a nonrelativistic effective theory arises once the antiparticles corresponding to a heavy stable particle are integrated out. We do so using a toy model of a single massive scalar field, and we work in position space to facilitate the identification of the couplings to gravitational fields.
Consider, then, a complex massive scalar field (we take a complex field to ensure lowenergy conservation of heavyparticle number) having action
$$\begin{array}{c}\frac{\mathcal{\mathcal{L}}}{\sqrt{g}}={g}^{\mu \nu}{\partial}_{\mu}{\phi}^{*}{\partial}_{\mu}\phi +{m}_{\phi}^{2}{\phi}^{*}\phi ,\end{array}$$ 
(36)

for which the conserved current for heavyparticle number is
$$\begin{array}{c}{J}^{\mu}=i{g}^{\mu \nu}({\phi}^{*}{\partial}_{\nu}\phi {\partial}_{\nu}{\phi}^{*}\phi ).\end{array}$$ 
(37)

Our interest is in exhibiting the leading couplings of this field to gravity, organized in inverse powers of
${m}_{\phi}$
. We imagine therefore a family of observers relative to whom the heavy particles move nonrelativistically, and whose foliation of spacetime allows the metric to be written as
$$\begin{array}{c}d{s}^{2}=(1+2\phi )d{t}^{2}+2{N}_{i}dtd{x}^{i}+{\gamma}_{ij}d{x}^{i}d{x}^{j},\end{array}$$ 
(38)

where
$i=1,2,3$
labels coordinates along the spatial slices which these observers define.
When treating nonrelativistic particles it is convenient to remove the rest mass of the heavy particle from the energy, since (by assumption) this energy is not available to other particles in the lowenergy theory. For the observers just described this can be done by extracting a timedependent phase from the heavyparticle field according to
$\phi \left(x\right)=F{e}^{i{m}_{\phi}t}\chi \left(x\right)$
.
$F=(2{m}_{\phi}{)}^{1/2}$
is chosen for later convenience, to ensure a conventional normalization for the field
$\chi $
. With this choice we have
${\partial}_{t}\phi =F({\partial}_{t}i{m}_{\phi})\chi $
, and the extra
${m}_{\phi}$
dependence introduced this way has the effect of making the large
${m}_{\phi}$
limit of the positivefrequency part of a relativistic action easier to follow.
With these variables the action for the scalar field becomes
$$\begin{array}{c}\frac{\mathcal{\mathcal{L}}}{\sqrt{g}}=\frac{{m}_{\phi}}{2}\left({g}^{tt}+1\right){\chi}^{*}\chi +\frac{i}{2}{g}^{t\mu}\left({\chi}^{*}{\partial}_{\mu}\chi {\partial}_{\mu}{\chi}^{*}\chi \right)+\frac{1}{2{m}_{\phi}}{g}^{\mu \nu}{\partial}_{\mu}{\chi}^{*}{\partial}_{\nu}\chi ,\end{array}$$ 
(39)

and the conserved current for heavyparticle number becomes
$$\begin{array}{c}{J}^{\mu}={g}^{\mu t}{\chi}^{*}\chi \frac{i}{2{m}_{\phi}}{g}^{\mu \nu}({\chi}^{*}{\partial}_{\nu}\chi {\partial}_{\nu}{\chi}^{*}\chi ).\end{array}$$ 
(40)

Here
${g}^{tt}=1/D$
,
${g}^{ti}={N}^{i}/D$
, and
${g}^{ij}={\gamma}^{ij}{N}^{i}{N}^{j}/D$
, with
${N}^{i}={\gamma}^{ij}{N}_{j}$
and
$D=1+2\phi +{\gamma}^{ij}{N}_{i}{N}_{j}$
.
Notice that for Minkowski space, where
${g}^{\mu \nu}={\eta}^{\mu \nu}=diag(,+,+,+)$
, the first term in
$\mathcal{\mathcal{L}}$
vanishes, leaving a result which is finite in the
${m}_{\phi}\to \infty $
limit. Furthermore – keeping in mind that the leading time and space derivatives are of the same order of magnitude (
${\partial}_{t}\sim {\nabla}^{2}/{m}_{\phi}$
) – the leading large
${m}_{\phi}$
part of
$\mathcal{\mathcal{L}}$
is equivalent to the usual nonrelativistic Schrödinger Lagrangian density,
${\mathcal{\mathcal{L}}}_{sch}={\chi}^{*}\left[i{\partial}_{t}+{\nabla}^{2}/\left(2{m}_{\phi}\right)\right]\chi $
. In the same limit the density of
$\chi $
particles also takes the standard Schrödinger form
$\rho ={J}^{t}={\chi}^{*}\chi +\mathcal{O}(1/{m}_{\phi})$
.
The next step consists of integrating out the antiparticles, which (by assumption) cannot be produced by the lowenergy physics of interest. In principle, this can be done by splitting the relativistic field
$\chi $
into its positiveand negativefrequency parts
${\chi}_{(\pm )}$
, and performing the functional integral over the negativefrequency part
${\chi}_{()}$
. (To leading order this often simply corresponds to setting the negativefrequency part to zero.) Once this has been done the fields describing the heavy particles have the nonrelativistic expansion
$$\begin{array}{c}{\chi}_{(+)}\left(x\right)=\int {d}^{3}p{a}_{p}{u}_{p}^{(+)}\left(x\right),\end{array}$$ 
(41)

with no antiparticle term involving
${a}_{p}^{*}$
. It is this step which ensures the absence of virtual heavyparticle loops in the graphical expansion of amplitudes in the lowenergy effective theory.
Writing the heavyparticle action in this way extends the standard parameterized postNewtonian (PPN) expansion [
68,
66,
67,
146]
to the effective quantum theory, and so forms the natural setting for an allorders powercounting analysis which keeps track of both quantum and relativistic effects.
For instance, for weak gravitational fields having
$\phi \sim {N}^{2}\sim {\gamma}_{ij}{\delta}_{ij}\ll 1$
, the leading gravitational coupling for large
${m}_{\phi}$
may be read off from Equation ( 39 ) to be
$$\begin{array}{c}{\mathcal{\mathcal{L}}}_{0}\approx \frac{{m}_{\phi}}{2}\sqrt{g}\left({g}^{tt}+1\right){\chi}^{*}\chi \approx {m}_{\phi}\left(\phi +\frac{{N}^{2}}{2}\right){\chi}^{*}\chi ,\end{array}$$ 
(42)

which for
${N}_{i}=0$
reproduces the usual Newtonian coupling of the potential
$\phi $
to the nonrelativistic mass distribution. For several
$\chi $
particles prepared in position eigenstates we are led in this way to considering the gravitational field of a collection of classical point sources.
The real power of the effective theory lies in identifying the subdominant contributions in powers of
$1/{m}_{\phi}$
, however, and the above discussion shows that different components of the metric couple to matter at different orders in this small quantity. Once
$\phi $
is shifted by the static nonrelativistic Newtonian potential, however, the remaining contributions are seen to couple with a strength which is suppressed by negative powers of
${m}_{\phi}$
, rather than positive powers.
A full powercounting analysis using such an effective theory, along the lines of the analogous electromagnetic problems [
34,
111,
103,
104,
127,
110]
, would be very instructive.
3.3 Effective field theory in curved space
All of the explicit calculations of the previous sections are performed for weak gravitational fields, which are well described as perturbations about flat space. This has the great virtue of being sufficiently simple to make explicit calculations possible, including the widespread use of momentumspace techniques. Much less is known in detail about effective field theory in more general curved spaces, although its validity is implicitly assumed by the many extant calculation of quantum effects in curved space [
19]
, including the famous calculation of Hawking radiation [
87,
88]
by black holes. This section provides a brief sketch of some effectivefield theory issues which arise in curvedspace applications.
For quantum systems in curved space we are still interested in the gravitational action, Equation (
28 ), possibly supplemented by a matter action such as that of Equation ( 36 ). As before, quantization is performed by splitting the metric into a classical background
${\hat{g}}_{\mu \nu}$
and a quantum fluctuation
${h}_{\mu \nu}$
according to
${g}_{\mu \nu}={\hat{g}}_{\mu \nu}+{h}_{\mu \nu}$
. A similar expansion may be required for the matter fields
$\phi =\hat{\phi}+\delta \phi $
, if these acquire vacuum expectation values
$\hat{\phi}$
.
The main difference from previous sections is that
${\hat{g}}_{\mu \nu}$
is not assumed to be the Minkowski metric. Typically we imagine the spacetime may be foliated by a set of observers, as in Equation ( 38 ),
$$\begin{array}{c}d{s}^{2}=(1+2\phi )d{t}^{2}+2{N}_{i}dtd{x}^{i}+{\gamma}_{ij}d{x}^{i}d{x}^{j}.\end{array}$$ 
(43)

and we imagine that the slices of constant
$t$
are Cauchy surfaces, in the sense that the future evolution of the fields is uniquely specified by initial data on a constant
$t$
slice. The validity of this initialvalue problem may also require boundary conditions for the fluctuations
${h}_{\mu \nu}$
such as that they vanish at spatial infinity.
3.3.1 When should effective Lagrangians work?
Many of the general issues which arise in this problem are similar to those which arise outside of gravitational physics when quantum fields are considered in the presence of background classical scalar or electromagnetic fields. In particular, there is a qualitative difference between the cases where the background fields are timeindependent or timedependent. The purpose of this section is to argue that we expect an effective field theory to work provided that the background fields vary sufficiently slowly with respect to time, an argument which in its relativistic context is called the `nice slice' criterion [
130]
.
Timeindependent backgrounds
In flat space, background fields which are timetranslation invariant allow the construction of a conserved energy
$H$
, which evolves forward the fields from one
$t$
slice to another. If
$H$
is bounded below^{
$\text{3}$
}
, then a stable ground state for the quantum fields (the `vacuum') typically exists, about which small fluctuations can be explored perturbatively. Since energy can be defined and is conserved, it can be used as a criterion for defining an effective theory which distinguishes between states which are `lowenergy' or `highenergy' as measured using
$H$
. Once this is possible, a lowenergy effective theory may be defined, and we expect the general uncertaintyprinciple arguments given earlier to ensure that it is local in time.
A similar statement holds for background gravitational fields, with a conserved Hamiltonian following from the existence of a timelike Killing vector field
${K}^{\mu}$
for the background metric
${\hat{g}}_{\mu \nu}$
^{
$\text{4}$
}
. For matter fields
$H$
is defined in terms of
${K}^{\mu}$
and the stress tensor
${T}_{\mu \nu}$
by the integral
$$\begin{array}{c}{H}_{mat}={\int}_{t}d{\Sigma}_{\mu}{K}^{\nu}{T}_{\nu}^{\mu}.\end{array}$$ 
(44)

Here the integral is over any spacelike slice whose measure is
$d{\Sigma}_{\mu}$
. Given appropriate boundary conditions this definition can also be extended to the fluctuations
${h}_{\mu \nu}$
of the gravitational field [
10,
11,
12,
6,
16,
8,
7,
13,
14,
15,
1]
.
If the timelike Killing vector
${K}^{\mu}$
is hypersurface orthogonal, then the spacetime is called static, and it is possible to adapt our coordinates so that
${K}^{\mu}{\partial}_{\mu}={\partial}_{t}$
and so the metric of Equation ( 43 ) specializes to
${N}_{i}=0$
, with
$\phi $
and
${\gamma}_{ij}$
independent of
$t$
. In this case we call the observers corresponding to these coordinates `Killing' observers. In order for these observers to describe physics everywhere, it is implicit that the timelike Killing vector
${K}^{\mu}$
be globally defined throughout the spacetime of interest.
If
$H$
is defined and is bounded from below, its lowest eigenstate defines a stable vacuum and allows the creation of a Fock space of fluctuations about this vacuum. As was true for nongravitational background fields, in such a case we might again expect to be able to define an effective theory, using
$H$
to distinguish `lowenergy' from `highenergy' fluctuations about any given vacuum.
It is often true that
${K}^{\mu}$
is not unique because there is more than one globallydefined timelike Killing vector in a given spacetime. For instance, this occurs in flat space where different inertial observers can be rotated, translated, or boosted relative to one another. In this case the Fock space of states to which each of the observers is led are typically all unitarily equivalent to one another, and so each observer has an equivalent description of the physics of the system.
An important complication to this picture arises when the timelike Killing vector field cannot be globally defined throughout all of the spacetime. In this case horizons exist, which divide the regions having timelike Killing vectors from those which do not. Often the Killing vector of interest becomes null at the boundaries of these regions. Examples of this type include the accelerated `Rindler' observers of flat space, as well as the static observers outside of a black hole. In this case it is impossible to foliate the entire spacetime using static slices which are adapted to the Killing observer, and the above construction must be reconsidered.
Putting the case of horizons aside for the moment, we expect that a sensible lowenergy/highenergy split should be possible if the background spacetime is everywhere static, and if the conserved energy
$H$
is bounded from below.
Timedependent backgrounds
In nongravitational problems, timedependence of the background fields need not completely destroy the utility of a Hamiltonian or of a ground state, provided that this time dependence is sufficiently weak as to be treated adiabatically. In this case the lowest eigenstate of
$H\left(t\right)$
for any given time defines both an approximate ground state and an energy in terms of which lowenergy and highenergy states can be defined (such as by using an appropriate eigenvalue
$\Lambda \left(t\right)$
of
$H\left(t\right)$
).
Once the system is partitioned in this way into lowenergy and highenergy state, one can ask whether a purely lowenergy description of time evolution is possible using only a lowenergy, local effective Lagrangian. The main danger which arises with timedependent backgrounds is that the time evolution of the system need not keep lowenergy states at low energies, or highenergy states at high energies. There are several ways in which this might happen:

∙
The time dependence of the background can be rapid, for instance having Fourier components which are larger than the dividing line
$\Lambda \left(t\right)$
between lowenergy and highenergy states. If so, the background evolution is not sufficiently adiabatic to prevent the direct production of `heavy' particles, and an effective field theory need not exist. The criterion for this not to happen would be that the energies of any produced particles be smaller than the dividing eigenvalue
$\Lambda \left(t\right)$
, which typically requires that
$\Lambda \left(t\right)$
be much smaller than the background time dependence, for instance if
$\Lambda \ll \dot{\Lambda}/\Lambda $
.

∙
Even for slowlyvarying backgrounds there can be other dangers, such as the danger of levelcrossing. For instance the dividing eigenvalue
$\Lambda \left(t\right)$
may slowly evolve to smaller values and so eventually invalidate an expansion in inverse powers of
$\Lambda \left(t\right)$
. In this case states which are prepared in the lowenergy regime may evolve out of it, leading to a breakdown in the lowenergy approximation. For instance, this might happen if
$\Lambda \left(t\right)$
were chosen to be the mass of an initially heavy particle, which is integrated out to obtain the effective theory.
If evolution of the background fields allows this mass to evolve to become too low, then eventually it becomes a bad approximation to have integrated it out.

∙
A related potential problem can arise if states whose energy is initially larger than
$\Lambda \left(t\right)$
enter the lowenergy theory by evolving into states having energy lower than
$\Lambda \left(t\right)$
^{
$\text{5}$
}
. This usually is only a problem for the effectivetheory formulation if the states which enter in this way are not in their ground state when they do so, since then new physical excitations would arise at low energies. Conversely, they are not a problem for the effective field theory description so long as they enter in their ground states, as can usually be ensured if the background time evolution is adiabatic.
What emerges from this is that an effective field theory can make sense despite the presence of timedependent backgrounds, provided one can focus on the evolution of lowenergy states (
$E<\Lambda \left(t\right)$
) without worrying about losing probability into highenergy states (
$E>\Lambda \left(t\right)$
). This is usually ensured if the background time evolution is sufficiently adiabatic.
A similar story should also hold for background spacetimes which are not globally static, but for which a globallydefined timelike hypersurfaceorthogonal vector field
${V}^{\mu}$
exists. For such a spacetime the observers for whom
${V}^{\mu}$
is tangent to worldlines can define a foliation of spacetime, as in Equation ( 43 ), but with the various metric components not being
$t$
independent. In this case the quantity defined by Equation ( 44 ) need not be conserved,
$H=H\left(t\right)$
, for these observers. A lowenergy effective theory should nonetheless be possible, provided
$H\left(t\right)$
is bounded from below and is sufficiently slowly varying (in the senses described above). If such a foliation of spacetime exists, following [
130]
we call it a `nice slice'.
3.3.2 General power counting
Given an effective field theory, the next question is to analyze systematically how small energy ratios arise within perturbation theory. Since the key powercounting arguments of the previous sections were given in momentum space, a natural question is to ask how much of the previous discussion need apply to quantum fluctuations about more general curved spaces. In particular, does the argument that shows how large quantum effects are at arbitraryloop order apply more generally to quantum field theory in curved space?
An estimate of higherloop contributions performed in position space is required in order to properly apply the previous arguments to more general settings. Such a calculation is possible because the crucial part of the earlier estimate relied on an estimate for the highenergy dependence of a generic Feynman graph. This estimate was possible on dimensional grounds given the highenergy behaviour of the relevant vertices and propagators. The analogous computation in position space is also possible, where it instead relies on the
operator product expansion [
156,
41]
. In position space one's interest is in how effective amplitudes behave in the shortdistance regime, rather than the limit of high energy. But the shortdistance limit of propagators and vertices are equally wellknown, and resemble the shortdistance limit which is obtained on flat space.
Consequently general statements can be made about the contributions to the lowenergy effective theory.