Matter Contributions to Gravitational Matching

Matter matching asks what remains in the gravitational EFT after a specified matter field is integrated out. A matter determinant contains closed matter loops with any number of external background gravitons, but no internal graviton or ghost propagator. Its low-momentum expansion shifts gravitational Wilson coefficients; using those shifts and retaining the same heavy loop explicitly would count the same physics twice.

Required background. One-Loop Graviton EFT distinguishes metric–ghost loops from matter loops; Renormalization of Gravitational Couplings by Matter Loops supplies the matter counterterms; and Matching Conditions Beyond Tree Level supplies loop-level matching.

Helpful background. Decoupling Theorems and Threshold Corrections fixes the low-energy expansion, while Heat Kernels and the Schwinger–DeWitt Expansion supplies its local coefficients.

Which internal lines have been integrated out

For a real scalar $\Phi$ with

$$P_\Phi=\Box+m^2+\xi R, \qquad \xi_{\mathrm{conf}}=-\frac16,$$

integrating over $\Phi$ at fixed $g_{\mu\nu}$ gives

$$\Gamma_\Phi[g] =\frac{i}{2}\operatorname{Tr}\log P_\Phi .$$

Expanding $\Gamma_\Phi$ in $h_{\mu\nu}$ generates one closed scalar loop with external gravitons. It does not generate a loop containing an internal graviton, because $g_{\mu\nu}$ has not been integrated over. A later path integral over low-energy metric fluctuations uses the matched gravitational action and separately creates graviton and ghost loops. Mixed heavy-scalar–graviton graphs require their own matching calculation at the corresponding higher loop order.

For momenta and curvature scales small compared with $m$, the scalar determinant has a local asymptotic expansion. In the site curvature and coupling convention, the curvature-squared heat-kernel density for a boundaryless four-dimensional region may be organized as

$$a_2^{(\Phi)} =\frac{1}{180} \left( R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -R_{\mu\nu}R^{\mu\nu} \right) +\frac12\left(\xi+\frac16\right)^2R^2 +\nabla_\mu V^\mu .$$

The last term is locally scheme dependent and becomes consequential with a boundary. The replacement $\xi_{\mathrm{standard}}=-\xi$ explains why the conformal combination is $\xi+1/6$ here. Parker and Toms derive the scalar coefficients and their convention translations in Parker and Toms 2009, §3.6, pp. 168–176.

The same determinant also contains its lower heat-kernel orders. The $a_0$ term is proportional to $m^4$ and threshold-matches the vacuum-energy coupling, while $a_1$ contains $m^2R$ and threshold-matches $M_{\mathrm{Pl}}^2$ in the declared action convention. The worked example below deliberately isolates $a_2$ because its task is curvature-squared matching; it does not imply that heavy matter leaves the zero- and two-derivative gravitational couplings unchanged.

The structure map sends this heat-kernel information into matching before any low-energy metric loop is evaluated.

Integrating out a heavy scalar produces matter-loop threshold terms in gravitational Wilson coefficients; subsequent metric and ghost loops are distinct contributions. The map is schematic and not to scale.

First application: one heavy scalar

Choose the local basis $\mathcal O_i=\{R_{\mu\nu\rho\sigma}^2,R_{\mu\nu}^2,R^2\}$ and define $b_i^{(\Phi)}$ by

$$\Gamma_\Phi^{\mathrm{local}} \supset \frac{1}{2(4\pi)^2} \log\!\left(\frac{m^2}{\mu^2}\right) \int\mathrm d^4x\sqrt{-g}\, \sum_i b_i^{(\Phi)}\mathcal O_i .$$

The displayed $a_2^{(\Phi)}$ gives

$$\left(b_{\mathrm{Riem}^2}^{(\Phi)}, b_{\mathrm{Ric}^2}^{(\Phi)}, b_{R^2}^{(\Phi)}\right) =\left( \frac1{180},-\frac1{180}, \frac12\left(\xi+\frac16\right)^2 \right)$$

up to the stated total derivative and basis changes. With the gravitational action written as $\sum_i c_i(\mu)\mathcal O_i$, equality of renormalized low-energy observables defines the threshold relation

$$c_i^{\mathrm{low}}(\mu) =c_i^{\mathrm{high}}(\mu) +\frac{b_i^{(\Phi)}}{2(4\pi)^2} \log\!\left(\frac{m^2}{\mu^2}\right) +\kappa_i .$$

Here $\kappa_i$ is a finite, scheme- and basis-dependent matching constant. Its sign is fixed only after the definitions of $c_i$ and the effective action are fixed as above. Choosing $\mu\simeq m$ avoids a large logarithm; running above and below the threshold then resums scale separation. Appelquist and Carazzone establish the low-energy expansion under its renormalizable-theory hypotheses in Appelquist and Carazzone 1975, §III, pp. 2862–2866.

To test the same coefficients in an observable, couple the metric to light external matter and consider momentum transfer $\lvert q^2\rvert\ll m^2$. The heavy scalar contribution to the graviton two-point function is analytic,

$$\Pi_{\mu\nu\rho\sigma}^{(\Phi)}(q) =q^4\sum_i b_i^{(\Phi)} \mathcal P_{\mu\nu\rho\sigma}^{(i)} \left[ \frac{1}{2(4\pi)^2} \log\!\left(\frac{m^2}{\mu^2}\right) +\text{finite} \right] +O\!\left(\frac{q^6}{m^2}\right),$$

where the $\mathcal P^{(i)}$ are the tensor projectors obtained by expanding $\mathcal O_i$. Inserting the matched $c_i^{\mathrm{low}}$ in the low-energy graviton exchange reproduces the full-theory amplitude through $q^4$. Field redefinitions may trade Ricci operators for light-matter contact terms; both must be transformed before comparing amplitudes.

Double-counting test

Suppose the low-energy calculation uses $c_i^{\mathrm{low}}$ above and also draws the heavy-scalar polarization loop. The $q^4\log(m^2/\mu^2)$ contribution then appears twice. The repair is categorical: either retain $\Phi$ as an active field and use high-energy coefficients, or remove it from propagating loops and use the low-energy matched coefficients. Near threshold, where a local $q^2/m^2$ expansion is inaccurate, one retains the full form factor or matches to an EFT with appropriate active modes.

This bookkeeping does not say that every matter effect is analytic. A light or massless field has nonanalytic cuts and remains active; its long-distance form factor cannot be absorbed into local $c_i$. Nor does the calculation include pure metric or ghost loops, which are separately counted on the preceding page.

Domain and failure conditions

The chapter comparison table requires the integrated species, mass and curvature hierarchy, boundary conditions, basis, subtraction scheme, active low-energy fields, and matched observable. If $\mathcal R$ denotes the largest independent curvature scale measured in the apparatus-defined orthonormal frame, the local result requires $\lvert q^2\rvert/m^2\ll1$ and $\mathcal R/m^2\ll1$; threshold and nonlocal regimes require the unexpanded form factor.

The failure map identifies duplicated heavy loops as a matching error, not an uncertainty estimate.

Heavy matter is either active in explicit loops or integrated into low-energy coefficients at a stated matching scale; mixing those descriptions without subtraction double counts the threshold. The map is schematic and not to scale.

References

• Appelquist, T., and J. Carazzone. “Infrared Singularities and Massive Fields.” Physical Review D 11, 2856–2861 (1975). doi:10.1103/PhysRevD.11.2856
• Parker, L., and D. Toms. Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge: Cambridge University Press, 2009. doi:10.1017/CBO9780511813924