Non-Local Dimension 6 Operators

• Non-local dimension 6 operators are effective interactions generated by graviton exchanges, characterized by a 1/Mₚ² suppression and non-local inverse d’Alembertian kernels.
• They exhibit a structure derived from bilinear energy–momentum tensor insertions, with loop corrections adding logarithmic modifications to the kernel.
• Phenomenologically, these operators form the gravitational portal for ultralight scalar dark matter, leading to distinctive oscillatory signals in precision experiments.

Non-local dimension 6 operators are effective interactions generated at leading order by quantized gravity in the perturbative (weak-field) regime and mediate couplings between the Standard Model (SM) and hidden sector fields. They are characterized by their non-local structure, dependence on inverse powers of the Planck mass $M_P$, and mass dimension six. At tree level, these operators systematically arise from the exchange of a graviton propagator connecting two energy–momentum tensor insertions, inducing effective non-local interactions of the schematic form $O_{\text{SM}}(x)\,F(x-y;M_P)\,O_{\text{hidden}}(y)$ with the non-local kernel $F(x-y;M_P)$. Perturbative quantum gravity neither generates dimension-5 operators nor local contact terms at this order. Phenomenologically, these operators govern the leading gravitational portal between SM fields and candidates for ultralight scalar dark matter at energies $E\ll M_P$ (Antonelli et al., 23 Dec 2025).

1. Derivation and Structure of Non-local Dimension 6 Operators

When two sectors are coupled only via gravity, the leading effective operators are obtained by integrating out the graviton in the linearized weak field expansion. The relevant interaction at leading order arises from a single graviton propagator connecting two energy–momentum tensors $T^{\mu\nu}$. The matter action expansion yields the coupling

$$S_{\rm matter}[g,\chi] = S_{\rm matter}[\eta,\chi] - \frac{1}{2M_P}\int d^4x\,h_{\mu\nu}(x)\,T^{\mu\nu}(x)$$

about $g_{\mu\nu} = \eta_{\mu\nu} + (1/M_P)h_{\mu\nu}$.

Gluing two such vertices via the graviton propagator in de Donder gauge\vspace{-5pt}

$$\langle 0|\,T\,h_{\mu\nu}(x)\,h_{\rho\sigma}(y)\,|0\rangle = 2i\,{\cal P}_{\mu\nu\rho\sigma}\,D(x-y)$$

(where $$D(x-y)=\int\frac{d^4q}{(2\pi)^4}\frac{-1}{q^2+i\epsilon}e^{iq\cdot(x-y)}$$) and integrating out $h_{\mu\nu}$, one obtains the non-local effective action

$$S_{\rm eff}^{\rm tree} = \frac{1}{4M_P^2} \int d^4x\,d^4y \Bigl[2T^{\mu\nu}(x)\frac{1}{\Box}T_{\mu\nu}(y) - T(x)\frac{1}{\Box}T(y)\Bigr].$$

Each $T^{\mu\nu}$ is expressed in terms of scalars, fermions, or vectors, yielding a non-local operator basis with explicit $1/M_P^2$ suppression and mass dimension six.

2. Operator Basis and Non-local Kernel

All tree-level operators fit the general form

$$\int d^4x\,d^4y\; O_{\rm SM}(x)\; F(x-y;M_P)\; O_{\rm hidden}(y)$$

where

$$F(x-y;M_P) = \frac{1}{M_P^2}\,D(x-y), \qquad D(x-y) = \int\frac{d^4p}{(2\pi)^4}\frac{-e^{ip\cdot(x-y)}}{p^2+i\varepsilon}.$$

In momentum space,

$F(p^2;M_P) = -\frac{1}{M_P^2(p^2 + i\varepsilon)}.$

Explicit operator examples include, for scalar fields,

$$O_{\phi,4} = \partial^\mu\phi(x)\partial^\nu\phi(x) \frac{1}{\Box} \partial_\mu\phi(y)\partial_\nu\phi(y)$$

and similar expressions for fermion and gauge fields. The operator basis (see Eqs. (7)–(9) in (Antonelli et al., 23 Dec 2025)) contains all allowed quadratic couplings through energy–momentum tensor bilinears, manifestly non-local due to the inverse d’Alembertian kernel.

3. Loop Corrections and Higher-order Effects

At one-loop, graviton self-interactions induce higher-curvature terms in the effective action, introducing terms such as

$$S = \int d^4x\sqrt{-g}\left[-\frac{1}{2}M_P^2 R + c_1R^2 + c_2 R_{\mu\nu}R^{\mu\nu} + b_1 R\ln\bigg(\frac{\Box}{\mu^2}\bigg)R + b_2 R_{\mu\nu}\ln\bigg(\frac{\Box}{\mu^2}\bigg)R^{\mu\nu}+\dots\right].$$

When linearized and integrated out, these generate higher-dimension contact terms suppressed by $1/M_P^4$, and yield logarithmic form factors modifying the $1/\Box$ kernel: $$\frac{1}{\Box} \rightarrow \frac{1}{\Box} + \frac{b_{\rm eff}}{M_P^2} \frac{\ln(\Box/\mu^2)}{\Box} + \mathcal{O}(M_P^{-4}).$$ Consequently, Wilson coefficients of the dimension-6 operators receive $O(1/M_P^4)\times \ln(-p^2/\mu^2)$ corrections. Genuine new local operators arise only at dimension eight, while dimension-6 structures remain non-local at this order.

4. Absence of Dimension-5 Operators and Selection Rules

Dimension-5 operators of the form

$\frac{1}{M_P}\,\phi\,O_{\rm SM}^{(4)}(x)$

are absent in the perturbative regime, as all graviton-mediated couplings require two insertions of $T^{\mu\nu}$, each with mass dimension four. The resulting structure always generates operators of dimension $\geq6$, arising as

$\frac{T(x)T(y)}{M_P^2\,\Box}$

and precluding any $1/M_P$-suppressed (dimension-5) portals. Thus, no single-graviton exchange can link a one-point insertion of a hidden scalar to a dimension-4 SM operator.

Non-perturbative, strong-field effects such as virtual Planck-mass black holes can in principle generate dimension-5 portals, as they are not bound by weak-field selection rules. An effective operator like $\frac{1}{M_P}\phi F_{\mu\nu}F^{\mu\nu}$ can arise from such non-perturbative dynamics, but would signal physics beyond the perturbative effective theory.

5. Phenomenological Implications and Ultralight Scalar Dark Matter

At energies $E\ll M_P$, the non-local kernel $F(p^2)\sim -1/(M_P^2\,p^2)$ leads to long-range $1/r^2$-type potentials between stress-energy sources. In laboratory and accelerator experiments, effects are suppressed by $1/M_P^2$ and by the squared momentum transfer. Sensitivity may be achieved via precision experiments such as atomic clocks, resonant bar detectors, or magnetometers, where oscillatory signals from hidden-sector fields can be probed.

For ultralight scalar dark matter of mass $m_\phi\sim10^{-22}-10^{-6}$ eV, the leading gravitationally induced portal is the dimension-6 non-local operator

$$\frac{1}{M_P^2} \partial^\mu\phi\partial_\nu\phi \frac{1}{\Box}F_{\mu\rho}F^{\nu\rho}.$$

On a classical, oscillating background $\phi(t) \simeq \phi_0\cos m_\phi t$, this induces

$$-\frac{\phi_0^2}{8M_P^2}\cos(2m_\phi t)F_{0\mu}F^{0\mu}$$

so the coupling to photons is quadratic in the field, oscillating at frequency $2m_\phi$ and suppressed by $\phi_0^2/M_P^2$, typically $\sim 10^{-62}$ for galactic dark-matter densities. No linear (dimension-5) $\phi F_{\mu\nu}F^{\mu\nu}$ coupling arises in perturbative gravity, enforcing that leading couplings are quadratic and non-local.

6. Summary and Theoretical Significance

In perturbative quantum gravity, the lowest-dimension induced operators coupling the SM to a hidden sector are dimension-6, non-local, and suppressed by $M_P^{-2}$. These operators are constructed from bilinears of $T^{\mu\nu}$, connected via an inverse-d’Alembertian kernel. Loop-induced corrections modify coefficients logarithmically, and true new local operators only occur at higher dimension. The absence of $1/M_P$-suppressed dimension-5 portals strictly follows from the structure of graviton–matter couplings. For ultralight scalar dark-matter models, the leading signatures in SM observables are quadratic, oscillating at twice the field frequency, with an amplitude set by $\phi_0^2/M_P^2$. Any observation of linear portals would point to either explicit non-gravitational ultraviolet couplings or non-perturbative quantum-gravity phenomena (Antonelli et al., 23 Dec 2025).