Power counting

3.2 Power counting

Since gravitons are weakly coupled, perturbative power-counting may be used to see how the high-energy scales $Mp$ 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: $gμν = ημν + hμν$ . 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 $hμν$ .

A power-counting estimate for the $L$ -loop contribution to the $ℰ$ -point graviton-scattering amplitude $𝒜 ℰ$ , 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 toy-model analysis is the existence for gravity of interactions involving two derivatives, which all come from the Einstein–Hilbert term in $ℒeff$ . (Such terms also arise for Goldstone bosons for symmetry-breaking patterns involving non-Abelian groups and are easily incorporated into the analysis.) The resulting estimate for $𝒜 ℰ$ turns out to be of order

where $∑ ′ Z = id Vid$ and $∑ ′ P = 2 + 2L + id(d − 2)Vid$ . 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 $Vik$ , Equation (31

) takes the equivalent form

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 low-energy approximations, $E ≪ Mp$ and $E ≪ m$ . 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 $𝒜 ℰ$ unless contributions from at least curvature-cubed interactions are included (for which $d = 6$ ).

These expressions permit a determination of the dominant low-energy 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 low-energy dynamics of gravitational waves!

But the next-to-leading contributions are also quite interesting. These arise in one of two ways, either (i) $L = 1$ and $V = 0 id$ for any $d ⁄= 2$ , or (ii) $L = 0$ , $∑ V = 1 i i4$ , $V i2$ is arbitrary, and all other $V id$ vanish. That is, the next to leading contribution is obtained by computing the one-loop corrections using only Einstein gravity, or by working to tree level and including precisely one curvature-squared 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 $(E ∕Mp )2$ , and the one-loop contribution carries an additional factor of $(1∕4π )2$ .

For instance, for 2-body graviton scattering we have $ℰ = 4$ , and so the above arguments imply the leading behaviour is $𝒜4(E ) ∼ A2(E ∕Mp )2 + A4(E ∕Mp )4 + ...$ , where the numbers $A2$ and $A4$ 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 ]:

where $s$ , $t$ and $u$ are the usual Mandelstam variables, all of which are proportional to the square of the centre-of-mass energy $Ecm$ . Besides vindicating the power-counting expectation that $𝒜 ∼ (E ∕Mp )2$ to leading order, this example also shows that the potentially frame-dependent energy $E$ , which is relevant in the power-counting analysis, is in this case really the invariant centre-of-mass energy $Ecm$ .

The one-loop correction to this result has also been computed [63 ], and is infrared divergent. These infrared divergences cancel in the usual way with tree-level bremsstrahlung diagrams [143 ], leading to a finite result [61 ], which is suppressed as expected relative to the tree contribution by terms of order $2 (E ∕4πMp )$ , 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 non-relativistic matter than of relativistic matter. This section describes the reasons for this difference.

3.2.1.1 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 self-interact. The matter Lagrangian for such a system is then

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 graviton-matter vertices are counted together with those from the Einstein–Hilbert term when counting the vertices having precisely two derivatives with $Vi2$ .

Particular kinds of higher-derivative 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 $A ∕M μ p$ and its derivatives $∂ ∕m μ$ can be included, along the lines of

If the dimensionless constant $ckn$ in these expressions is $𝒪 (1)$ , then the power-counting result for the dependence of amplitudes on $m$ and $Mp$ is the same as it is for the pure-gravity theory, with vertices formed from $Δ ℒn$ counted amongst those with $d = (2 + 2k)n$ derivatives. If $m ≪ Mp$ it is more likely that powers of $Aμ$ come suppressed by inverse powers of $m$ rather than $Mp$ , however, in which case additional $A μ$ vertices are less suppressed than would be indicated above. The extension of the earlier power-counting estimate to this more general situation is straightforward.

Similar estimates also apply if a mass $m φ$ for the scalar field is included, provided that this mass is not larger than the energy flowing through the external lines: $m φ ≲ E$ . This kind of mass does not change the power-counting 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 infrared-finite quantities are at most logarithmically singular as $m φ → 0$ [149 ], and so their expansion in $m φ∕E$ simply adds terms for which factors of $E$ are replaced by smaller factors of $m φ$ . But the above discussion can change dramatically if $m φ ≫ E$ , 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 power-counting given above only directly applies to relativistic particles.

3.2.1.2 Non-relativistic matter

The situation is more complicated if the matter particles move non-relativistically, since in this case the particle mass is much larger than the momenta involved in the external lines, $p = |p| ≪ m φ$ , so $2 E ≈ m φ + p ∕(2m φ) + ...$ . 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 $Mp$ . Indeed, most of the tests of general relativity involve the gravitational interactions and orbits of very non-relativistic ‘particles’, like planets and stars. How can this be understood?

The case of non-relativistic 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. Non-relativistic 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 low-energy theory provided they are stable (or extremely long-lived), and so cannot decay and release enough energy to invalidate the low-energy approximation. Some well-known examples of this include the low-energy 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 heavy-quark effective theory (HQET) [90 , 91 , 108 ]), and the interactions of electrons and nuclei in atomic physics (as described by non-relativistic 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 anti-particles need not be included since they would have already been integrated out to obtain the effective field theory of interest. As a result heavy-particle 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 power-count 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 power-counting 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 heavy-particle momenta can be large, $p ≫ Mp$ , they only arise in physical quantities through the small relativistic parameter $p∕m φ ∼ v$ rather than through $p∕Mp$ , extending the suppression of quantum effects obtained earlier to non-relativistic problems.

Unfortunately, if a calculation is performed within a covariant gauge, individual Feynman graphs can depend on large powers like $m φ∕Mp$ , even though these all cancel in physical amplitudes. For this reason an all-orders inductive proof of the above power-counting remains elusive. As Donoghue and Torma [60 ] also make clear, progress towards such an all-orders power-counting result is likely to be easiest within a physical, non-covariant gauge, since such a gauge allows powers of small quantities like $v$ to be most easily followed.

3.2.1.3 Non-relativistic 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, $1∕Mp$ and $1∕m φ$ – 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 slowly-moving particles, Lorentz invariance is not simply realized on the corresponding heavy-particle fields. Second, since the effective theory does not contain antiparticles for the heavy particles, the heavy fields which describe them contain only positive-frequency parts. To illustrate how these features arise, we briefly sketch how such a non-relativistic 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 low-energy conservation of heavy-particle number) having action

for which the conserved current for heavy-particle number is

Our interest is in exhibiting the leading couplings of this field to gravity, organized in inverse powers of $m φ$ . We imagine therefore a family of observers relative to whom the heavy particles move non-relativistically, and whose foliation of spacetime allows the metric to be written as

where $i = 1, 2,3$ labels coordinates along the spatial slices which these observers define.

When treating non-relativistic 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 low-energy theory. For the observers just described this can be done by extracting a time-dependent phase from the heavy-particle field according to $−imφt φ(x) = F e χ(x)$ . $− 1∕2 F = (2m φ)$ is chosen for later convenience, to ensure a conventional normalization for the field $χ$ . With this choice we have $∂tφ = F (∂t − im φ )χ$ , and the extra $m φ$ -dependence introduced this way has the effect of making the large- $m φ$ limit of the positive-frequency part of a relativistic action easier to follow.

With these variables the action for the scalar field becomes

and the conserved current for heavy-particle number becomes

Here $gtt = − 1∕D$ , $gti = N i∕D$ , and $gij = γij − N iN j∕D$ , with $N i = γijNj$ and $D = 1 + 2φ + γijNiNj$ .

Notice that for Minkowski space, where $μν μν g = η = diag (− ,+, +, + )$ , the first term in $ℒ$ vanishes, leaving a result which is finite in the $m φ → ∞$ limit. Furthermore – keeping in mind that the leading time and space derivatives are of the same order of magnitude ( $∂t ∼ ∇2 ∕m φ$ ) – the leading large- $m φ$ part of $ℒ$ is equivalent to the usual non-relativistic Schrödinger Lagrangian density, $ℒsch = χ ∗[i∂t + ∇2 ∕(2m φ)]χ$ . In the same limit the density of $χ$ particles also takes the standard Schrödinger form $t ∗ ρ = J = χ χ + 𝒪 (1∕m φ)$ .

The next step consists of integrating out the anti-particles, which (by assumption) cannot be produced by the low-energy physics of interest. In principle, this can be done by splitting the relativistic field $χ$ into its positive- and negative-frequency parts $χ(±)$ , and performing the functional integral over the negative-frequency part $χ (−)$ . (To leading order this often simply corresponds to setting the negative-frequency part to zero.) Once this has been done the fields describing the heavy particles have the non-relativistic expansion

with no anti-particle term involving $a∗ p$ . It is this step which ensures the absence of virtual heavy-particle loops in the graphical expansion of amplitudes in the low-energy effective theory.

Writing the heavy-particle action in this way extends the standard parameterized post-Newtonian (PPN) expansion [68 , 66 , 67 , 146 ] to the effective quantum theory, and so forms the natural setting for an all-orders power-counting analysis which keeps track of both quantum and relativistic effects. For instance, for weak gravitational fields having $φ ∼ N 2 ∼ γij − δij ≪ 1$ , the leading gravitational coupling for large $m φ$ may be read off from Equation (39 ) to be

which for $Ni = 0$ reproduces the usual Newtonian coupling of the potential $φ$ to the non-relativistic mass distribution. For several $χ$ 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 $1∕m φ$ , however, and the above discussion shows that different components of the metric couple to matter at different orders in this small quantity. Once $φ$ is shifted by the static non-relativistic Newtonian potential, however, the remaining contributions are seen to couple with a strength which is suppressed by negative powers of $m φ$ , rather than positive powers. A full power-counting analysis using such an effective theory, along the lines of the analogous electromagnetic problems [34 , 111 , 103 , 104 , 127 , 110 ], would be very instructive.