Nonlocal Spin-2 Propagator Overview

• Nonlocal spin-2 propagators are defined by inverse d'Alembertian operators that modify gravitational fluctuations and soften the ultraviolet behavior.
• They ensure ghost-free dynamics by replacing problematic higher-derivative poles with a positive-residue massive spin-2 pole, as revealed by Barnes–Rivers projectors.
• The models produce finite corrections to gravitational potential, influencing black hole thermodynamics and regularizing singularities in quantum gravity frameworks.

A nonlocal spin-2 propagator arises in quantum-inspired modifications of gravity where nonlocal form factors replace or supplement the local curvature-squared terms of the Einstein–Hilbert action. These nonlocal operators alter the spectrum of gravitational fluctuations, regulate ultraviolet behavior, and potentially evade ghost instabilities endemic to higher-derivative local gravity theories. Precise realization of nonlocal spin-2 propagators depends on the choice of nonlocal kernel, symmetry constraints, and the desired pole structure in the graviton sector.

1. Nonlocal Quadratic Gravity: Framework and Action

Nonlocal quadratic gravity extends the Einstein–Hilbert action by curvature-squared operators “smeared” with inverse d'Alembertian kernels, introducing a nonlocality scale $\mu$ and loop-suppressed coupling $\alpha$ (Rocha, 29 Jan 2026). In $4$ dimensions with $8\pi G=1$, the action reads:

$$S = \int d^4x\,\sqrt{-g} \left\{ \frac{1}{2}R + \alpha \left[ R_{\mu\nu}(\Box-\mu^2)^{-1}R^{\mu\nu} - \frac{1}{4}R(\Box-\mu^2)^{-1}R \right] \right\}$$

Here, $(\Box-\mu^2)^{-1}$ nonlocally couples curvature invariants, with $\mu$ the mass scale generated by integrating out heavy degrees of freedom. This nonlocality softens the short-distance behavior and modifies the graviton propagator structure.

2. Linearized Spectrum and Spin-2 Projectors

To analyze the spectrum, one perturbs around flat Minkowski space: $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, with $|h_{\mu\nu}|\ll 1$, and expands the action to quadratic order. The resulting kinetic operator is typically decomposed using Barnes–Rivers spin projectors:

• $P^{(2)}$ isolates pure spin–2 transverse tensor components.
• $P^{(0)}$ isolates scalar (trace) components.

For the above nonlocal quadratic model, the scalar sector remains unaffected and propagates as in Einstein gravity: $\Pi_0(p^2) = 1/p^2$. The spin–2 sector propagator is

$$\Pi_2(p^2) = \frac{p^2+\mu^2}{p^2\left[ (1-\alpha)p^2 + \mu^2 \right]}$$

This propagator exhibits distinctive pole structure induced by nonlocality (Rocha, 29 Jan 2026).

3. Pole Structure and Ghost-Freedom

Detailed pole analysis reveals:

• Massless graviton pole: As $p^2\to 0$, $\Pi_2(p^2)$ reduces to the standard massless graviton propagator with positive residue, retaining unitarity.
• Massive spin–2 pole: A second pole arises at $p^2 = -m^2 = -\mu^2/(1-\alpha)$ ($m^2>0$ for $0<\alpha<1$), with positive residue $\alpha/(1-\alpha)$. This mode has positive norm and does not introduce ghosts at quadratic level.
• Removable singularity: The numerator vanishes at $p^2=-\mu^2$, but this is a removable singularity, not a physical pole.

The quadratic action for the spin–2 sector can be written in momentum space as

$$S_{spin-2}^{(2)} = \frac{1}{2}\int d^4p\,h^{(2)}(-p)\,P^{(2)}\,[p^2((1-\alpha)p^2+\mu^2)/(p^2+\mu^2)]\,h^{(2)}(p)$$

Near the massive pole, the kinetic term retains correct sign, confirming absence of ghost instabilities in this effective model (Rocha, 29 Jan 2026).

4. Nonlocal Form Factors and Generalized Propagators

The propagator’s structure is determined by nonlocal form factors $f(\Box)$, typically chosen to be entire functions. To ensure ghost-freedom and appropriate ultraviolet (UV) behavior, requirements include $f(0)=1$ and $f(z)>0$ for $z\in\mathbb{R}$ (Buoninfante et al., 2020, Abel et al., 2019).

An iterative family of nonlocal gravitational models is constructed using

$$f^{(n)}(z) = e^z\,z^{-n}[\,n! - n\,\Gamma(n,z)\,], \quad \text{with } \Gamma(n,z) = \int_z^\infty t^{n-1}e^{-t}dt$$

Each $f^{(n)}$ yields a ghost-free graviton propagator:

$$D_{μν,ρσ}^{(n)}(p) = \frac{1}{f^{(n)}(p^2)}\left[\frac{\mathcal{P}^2_{μν,ρσ}}{p^2} - \frac{\mathcal{P}^0_{s,μν,ρσ}}{2p^2} \right]$$

The only real pole is at $p^2=0$; any other poles are complex-conjugate and do not spoil tree-level unitarity (Buoninfante et al., 2020).

5. Worldline Inversion Symmetry and Unique Nonlocal Models

Worldline inversion symmetry, the particle analogue of modular invariance in string theory, provides a criterion for constructing unique nonlocal gravity theories (Abel et al., 2019). Imposing $t\rightarrow 1/(\mathcal{M}^4 t)$ invariance in the Schwinger proper-time representation, the form factor in the propagator is fixed as:

$K_1\left(2p^2/\mathcal{M}^2\right)$

with the explicit spin–2 propagator

$$\Pi_{μν,ρσ}(p) = \left(\eta_{μρ}\eta_{νσ}+\eta_{μσ}\eta_{νρ}-\eta_{μν}\eta_{\rho\sigma}\right)\frac{2}{\mathcal{M}^2}K_1\left(2p^2/\mathcal{M}^2\right)$$

Here $K_1$ is the modified Bessel function, an entire function with no zeros for $\Re(z^2)>0$, preventing extra poles besides $p^2=0$. The propagator thus remains ghost-free throughout and exponentially suppresses short-distance singularities in the deep UV ($p^2\gg\mathcal{M}^2$).

6. Gravitational Potential and Physical Consequences

Nonlocal effects introduce finite-range corrections to the classical gravitational potential via the massive spin–2 pole (Yukawa screening), shifting horizons and modifying black hole thermodynamics, including Bekenstein–Hawking entropy, Hawking temperature, and chemical potential. For large distances $r\gg\mu^{-1}$, nonlocality becomes irrelevant and GR is recovered; for $r\lesssim\mu^{-1}$, the additional pole softens the UV behavior, regularizing singularities such as the Coulomb potential near the origin (Rocha, 29 Jan 2026, Buoninfante et al., 2020).

A plausible implication is improved stability for small black holes, reduction in negative specific heat, and the absence of first-order phase transitions in the free energy landscape. Furthermore, nonlocal models admit “hairy” black hole solutions that are free of ghost instabilities at the quadratic and classical levels.

7. Comparison with Local Theories and Ultraviolet Properties

In local quadratic gravity, higher-derivative terms typically induce Ostrogradsky ghosts due to additional high-momentum poles with negative residues. By contrast, nonlocality replaces these problematic ghost poles with a massless graviton pole and either a single massive spin–2 pole with positive residue or, in generalized models, only complex-conjugate poles and no real ghosts (Rocha, 29 Jan 2026, Buoninfante et al., 2020).

At the technical level, when the nonlocality scale $\mu\rightarrow\infty$ (or $\Box\ll\mu^2$), nonlocal propagators reduce to their local counterparts:

$$(\Box-\mu^2)^{-1} = -\mu^{-2}\sum_{n\geq 0}(\Box/\mu^2)^n$$

yielding

$$S_{local} \sim -\alpha/\mu^2 (R_{\mu\nu}R^{\mu\nu} - \tfrac{1}{4}R^2) + \mathcal{O}(\mu^{-4})$$

with quadratic gravity recovered for $\alpha>0$. In nonlocal models with worldline inversion symmetry or generalized entire-based form factors, ultraviolet divergences are exponentially regulated, ensuring improved quantum behavior and finiteness in loop calculations.

Table: Propagator Structures in Nonlocal Gravity Theories

Model/Reference	Propagator Pole(s)	UV Behavior
Nonlocal quadratic (Rocha, 29 Jan 2026)	$p^2=0$ (massless), $p^2=-m^2$ (massive, $m^2>0$)	Yukawa screened, softened UV
Worldline inversion (Abel et al., 2019)	$p^2=0$ (massless)	Exponential damping ($\exp[-2p^2/\mathcal{M}^2]$)
Generalized entire (Buoninfante et al., 2020)	$p^2=0$, complex-conjugate (no real ghost)	Super-polynomial/exp. decay

In summary, nonlocal spin-2 propagators represent a robust theoretical mechanism to evade ghost instabilities, regularize ultraviolet behavior, and achieve novel black hole solutions with quantum-inspired features. The precise spectral content and physical consequences depend critically on the choice of nonlocal form factor, with worldline inversion symmetry and entire-function-based constructions providing systematic pathways to ghost-free, UV-finite gravity models (Rocha, 29 Jan 2026, Abel et al., 2019, Buoninfante et al., 2020).