Classical gravitational scattering with a massive scalar mediator

Abstract: We consider the classical scattering of two gravitating compact objects in the presence of a massive scalar mediator, providing a simple model of exotic phenomena. Through dimensional analysis, we argue that such a process can only be classical in the presence of gravity, a consequence of which is that perturbing in the coupling of the scalar to a worldline is not separate from the post-Minkowskian expansion. When computing asymptotic observables, the massive mediator complicates the Fourier transforms to impact-parameter space at next-to-leading order. We reduce these to univariate parametric integrals - amenable to numerical integration - and produce analytic results for the linear impulse and the scattering angle to the second post-Minkowskian order. The scattering angle exhibits a resonance when the range of the scalar-mediated force is comparable to the impact parameter, offering a distinctive signature of a massive mediator. In the opposite, large-mass regime we uncover a screening effect: the scalar cloud sourced by each compact object carries negative energy, reducing its gravitational mass by an amount linear in the scalar's mass. Both of these phenomena are next-to-leading-order effects.

• The paper computes the linear impulse and conservative scattering angle through second post-Minkowskian order, treating scalar and gravitational couplings at the same perturbative order.
• Mass-dependent Fourier transforms remain analytically controlled with at most one parametric integral, while the massive results require an ab initio treatment rather than a small-mass expansion.
• The paper finds an NLO resonance near μ|b|≈1 that can reach roughly 10% of the gravitational angle, alongside universal heavy-mediator screening δmᵢ=−cᵢ,₁²μ/(8π).

Overview and motivation

This paper studies the classical scattering of two gravitating compact objects that are also charged under a massive real scalar field, formulated within the worldline quantum field theory (WQFT) framework (2608.14267). The setup is motivated by scenarios in which scalar degrees of freedom accompany compact binaries — boson stars, ultralight scalar clouds from superradiance, or scalar dark matter — and the authors compute two key asymptotic observables, the linear impulse and the conservative scattering angle, through second post-Minkowskian (2PM) order. Their central technical achievements are (i) a correct organisation of the perturbative expansion in which scalar couplings count at the same PM order as gravitational ones, and (ii) compact analytic control of the massive-mediator Fourier transforms to impact-parameter space, leaving at most one unevaluated univariate parametric integral.

The bulk action augments the Einstein–Hilbert term in $d=4-2\varepsilon$ dimensions by a minimally coupled real scalar of mass $\mu$, and each worldline carries operators $c_{i,j}\phi(x_i)^j$ truncated at $j=2$, sufficient for the targeted precision.

Power counting: scalar couplings are not separate from the PM expansion

A key structural point is dimensional analysis performed without setting $\hbar=1$. Since $[\hbar]=[M][L]$, the natural quantum scalings $c_{i,j}\sim m_*^{1-j}\hbar^{j/2}$ have no classical analogue; instead, in the presence of gravity the authors adopt the classical scaling $c_{i,j}=\bar{c}_{i,j}\,m_i G^{j/2}$ with $\mu$ an independent inverse-length scale ($\mu\sim m_\phi G^k$ is impossible for any $\mu$0). This has the important consequence that replacing a graviton leg by a scalar leg does not change a vertex's PM order: the "purportedly 2PM" impulse of ref. Bhattacharyya et al. in fact mixes 2PM and 3PM contributions. Consequently, NLO accuracy requires diagrams with single-scalar exchange, mixed graviton–scalar exchange, and double-scalar exchange, organized into three non-mixing families: family 1 (one massive propagator), family 2a (two massive propagators, $\mu$1-dependent), and family 2b (two massive propagators, purely $\mu$2-dependent).

At leading order, the impulse is

$\mu$3

with $\mu$4; like-sign charges yield attraction, opposite signs repulsion. All dependence on $\mu$5 and $\mu$6 enters only through $\mu$7.

Fourier transforms with full mass dependence

The NLO impulse involves three novel Fourier transforms containing incomplete beta functions with mass-dependent arguments. The paper's method reduces each to a Hankel transform via the integral representation of $\mu$8 and a generalised Bessel-transform identity for $\mu$9, performing all but one parametric integral analytically. A physical consistency check — the on-shell constraint $c_{i,j}\phi(x_i)^j$0 — fixes the remaining integral in the combination appearing in family 2b, reducing it to $c_{i,j}\phi(x_i)^j$1, verified numerically.

Two structural observations follow. First, because modified Bessel functions decay exponentially, all families except 2a vanish as $c_{i,j}\phi(x_i)^j$2: the long-range effects decouple at large mediator mass, while family 2a grows with $c_{i,j}\phi(x_i)^j$3 for reasons explained below. Second, although the massless limit is smooth at both LO and NLO, the massive result cannot be reconstructed by expanding in small $c_{i,j}\phi(x_i)^j$4: already at tree level, any nonzero $c_{i,j}\phi(x_i)^j$5 admits a momentum region $c_{i,j}\phi(x_i)^j$6 where a small-mass expansion fails, producing a $c_{i,j}\phi(x_i)^j$7 rather than the required $c_{i,j}\phi(x_i)^j$8 structure. The massive case must be computed ab initio, though it admits the massless case as a limit.

Mass screening by the scalar cloud

In the $c_{i,j}\phi(x_i)^j$9 regime, the two scalar propagators of family 2a pinch onto the static worldline, generating an effective transverse graviton–worldline vertex that exactly satisfies the Ward identity. Matching this to the standard mass-monopole vertex yields an effective mass shift

$j=2$0

Crucially, the shift is negative regardless of the signs of the scalar charges: the scalar cloud sourced by each body invariably screens its gravitational mass. The screening magnitude is controlled by the ratio of the object's Schwarzschild radius to the scalar force range, $j=2$1, which may be arbitrary without leaving the PM domain $j=2$2. Consistency is demonstrated by showing that the $j=2$3 limit of the full 2PM impulse equals the correction obtained by inserting $j=2$4 into the 1PM gravitational impulse. The authors leave open whether this constitutes an observable effect or merely a finite renormalisation absorbed into the physical masses.

Scattering angle and resonance phenomenology

Using the conservative definition $j=2$5, the total angle splits as $j=2$6, with $j=2$7 the known Westpfahl result. Numerical exploration for equal masses, $j=2$8, and natural couplings $j=2$9 over the parameter space $\hbar=1$0 reveals:

Phenomenon	Regime	Origin
Resonance peak in $\hbar=1$1	$\hbar=1$2 ($\hbar=1$3)	Interplay of LO decay and NLO terms
Repulsive-case maximum	$\hbar=1$4	Sign-flipped charge $\hbar=1$5
Unbounded growth of $\hbar=1$6	$\hbar=1$7	Family-2a mass screening, not a long-range force

Bold claim: the scalar contribution can reach one tenth of the purely gravitational angle at its extremum, and this resonance emerges only once NLO effects are included — the LO result is monotonic in $\hbar=1$8. Because the resonance localises at impact parameters comparable to the Compton-like scale $\hbar=1$9, sampling over $[\hbar]=[M][L]$0 could in principle serve as a direct indicator of a nonzero scalar mass in scattering observations, though the authors caution that observability cannot be assessed from their analysis alone.

Limitations and open questions

The analysis is confined to a real scalar with minimal bulk interactions; solitonic potentials (quartic/sextic) relevant to realistic boson-star models first enter at two-loop order and are neglected, as are spin, tidal, and radiation-reaction effects beyond what the truncation captures. Complex scalars with harmonically varying worldline sources — the appropriate effective description for solitonic boson stars — introduce new scales and are deferred to future work. The naturalness assumption $[\hbar]=[M][L]$1 underlies the quantitative significance estimates, and the physical interpretation of the screening shift as renormalisation versus observable effect remains unresolved. Finally, the extension beyond 2PM would require higher-dimensional worldline operators and the corresponding multi-loop machinery.

Conclusion

The paper establishes analytic control of classical two-body scattering mediated jointly by gravitons and a massive scalar to 2PM order, clarifying that scalar and gravitational couplings must be counted together in the PM expansion and supplying compact, essentially closed-form Fourier transforms with full mass dependence. Two robust physical signatures emerge: a resonance in the scattering angle when the mediator range matches the impact parameter — a next-to-leading-order effect reaching roughly ten percent of the gravitational angle — and a universal negative mass screening $[\hbar]=[M][L]$2 in the heavy-mediator regime. Together these provide concrete handles for identifying massive scalar degrees of freedom in precision gravitational-scattering observables.