Resummation and Strong-Field Completion

A finite PN or PM expansion determines only a finite set of weak-field coefficients. EOB resummation rewrites those coefficients so that exact limiting structures—test-body horizons, tails, light rings, or known singular behavior—can remain visible at finite order. It can greatly improve a model, but it also selects a strong-field completion not uniquely implied by the input series. This page compares the principal choices and gives tests that separate coefficient matching from genuine predictive success.

Evidence cutoff. The assessment below includes literature and model information checked through 2026-08-12. Later analytic results, numerical-relativity releases, corrections, and independent reproductions lie outside this evidence window.

Required background. Presymplectic Potentials and Currents supplies canonical equivalence and Hamiltonian phase-space reasoning.

Helpful background. The Einstein Equation supplies the exact dynamics being approximated; Weak Metric Perturbations and Gauge Freedom supplies asymptotic matching; Black-Hole Perturbation Theory supplies strong-field test-mass benchmarks; and Relativistic Two-Body Dynamics supplies the PN, PM, and self-force domains.

What a resummation changes

Suppose an EOB potential is known only as

$$F_N(u)=\sum_{k=0}^{N}c_k u^k, \qquad u=\frac{GM}{r}.$$

Any function $\widetilde F(u)$ satisfying

$$\widetilde F(u)-F_N(u)=O(u^{N+1})$$

contains exactly the same information through the computed order. Infinitely many such functions exist, and they can differ substantially when $u$ is not small. Resummation is therefore a constrained modeling operation: preserve every known coefficient and exact limit, choose a representation expected to behave better, and then test it on data not used to choose it.

This distinction already appears in the original EOB construction, where a Padé form was used to preserve useful strong-field structure of the effective potential Buonanno and Damour 1999, §§ V–VII, PDF. The modern framework has more analytic input, but the logical point is unchanged. Re-expansion verifies consistency; it does not verify the continuation.

Two credible interpretations coexist.

Position A: physically informed continuation. Exact test-mass functions, known singular factors, tail logarithms, and self-force behavior reveal analytic structure that a raw Taylor polynomial obscures. A resummation that builds in those structures can use finite perturbative information more efficiently and improve predictions before calibration.

Position B: benchmark-defined model choice. Beyond the matched order, the continuation is not fixed by perturbation theory. Padé degree, logarithmic variable, factorization, or radial redefinition can create poles, branches, or apparently good behavior by accident. Reliability is established only on independent self-force, scattering, or numerical-relativity data in a declared domain.

These positions are not mutually exclusive. The first motivates candidate forms; the second states the evidence required to trust them.

Taylor, Padé, and logarithmic potentials

Taylor truncation is the transparent baseline. It introduces no denominator and makes PN order counting explicit. It is often adequate at small $u$, but a generic truncation does not automatically preserve the intended horizon zero and cannot reproduce a pole. A polynomial can preserve a known zero if that factor is built in; Schwarzschild’s $A=1-2u$ is itself polynomial. The Taylor seed should be included in a same-input comparison because improvement by resummation must be demonstrated on independent data.

A Padé approximant $P_m^n(u)=N_m(u)/D_n(u)$, with $m+n=N$, is chosen so that

$$\frac{N_m(u)}{D_n(u)} =F_N(u)+O(u^{N+1}).$$

The denominator can represent a pole or shift a zero using only a few coefficients. This is useful when the exact limit suggests rational structure. It is also hazardous: a zero of $D_n$ on the physical interval creates a spurious singularity. Different degrees $(m,n)$ share the same known series but may behave very differently. Current SEOBNRv5HM, for example, uses specified Padé orders for its nonspinning $A$ and $\bar D$ potentials rather than treating “Padé resummation” as a unique operation Pompili et al. 2023, § IV, Eqs. (63)–(64), PDF.

A logarithmic or exponential form first factors a known baseline and then exponentiates a residual series. Schematically,

$$F(u;\nu) =F_0(u) \exp\left[\nu f_1(u)+\nu^2f_2(u)+\cdots\right].$$

Matching fixes the Taylor coefficients of the $f_i$. This form can preserve positivity of the exponential factor and isolate a test-mass zero in $F_0$. Yet a truncated logarithm can still introduce unwanted zeros through its prefactor, a branch restriction if the logarithm is defined from a sign-changing function, or excessive growth outside the matching region. In SEOBNR development, changing from a logarithmic to selected Padé potential forms was tied to a simultaneous change in merger attachment and calibration, so the performance difference cannot be attributed to one algebraic step alone Pompili et al. 2023, § IV and Appendix E.

For every rational or logarithmic candidate, scan the full intended phase-space interval. Check denominator zeros, branch arguments, positivity of the radicand in $H_{\rm eff}$, existence and ordering of turning points, and smooth parameter dependence as $\nu$ and spins vary.

Factorized waveforms and centrifugal radius

Waveform resummation is naturally multiplicative. Define the mode-parity index by $\pi_{\ell m}=(\ell+m)\bmod 2$, with $\pi_{\ell m}=0$ for even-parity mass-type modes and $\pi_{\ell m}=1$ for odd-parity current-type modes. Then

$$H_{\ell m} =H_{\ell m}^{\rm N} \widehat S_{\rm eff}^{(\pi_{\ell m})} T_{\ell m}e^{i\delta_{\ell m}} \left(\rho_{\ell m}\right)^\ell N_{\ell m}.$$

The factors isolate a Newtonian mode, a strong-field source, leading tails, residual phase, residual amplitude, and nonquasicircular behavior. This prevents a large additive coefficient in one physical effect from contaminating all others. The $\ell$-th-root amplitude $\rho_{\ell m}$ is particularly useful because the unfactored residual coefficients tend to grow with multipole order. The construction and its test-mass comparisons are given in Damour, Iyer, and Nagar 2009, § II and §§ III–IV.

Factorization is still not unique. One may move a known term between the source and residual amplitude, Padé resum $\rho_{\ell m}$, or insert effective high-PN coefficients. Each variant must re-expand to the same known waveform and use the same phase and tail convention. A favorable flux comparison does not automatically validate the conservative Hamiltonian.

Spin-spin terms can be reorganized through a centrifugal radius. In the Kerr test-mass limit a characteristic form is

$$r_c^2=r^2+a^2+\frac{2GMa^2}{r},$$

where $a=S/M$ is the Kerr length parameter. Comparable-mass EOB models deform this expression so that powers of $1/r_c$ absorb selected even-in-spin couplings. This keeps exact Kerr structure visible and simplifies gyro functions. The deformation beyond known spin orders remains a completion choice. The 2025 LISA waveform review describes the centrifugal-radius and systematic Padé strategy in TEOBResumS, including the distinct GIOTTO and Dalí potential content LISA Consortium Waveform Working Group 2025, § 4.5.4 and Table 10, pp. 97–99.

Strong-field scattering as a counterexample

Scattering is a demanding test because a trajectory samples a wide radial interval and approaches the capture threshold. Swain, Pratten, and Schmidt compared three PM-informed resummations—the angular-momentum-singularity or $\mathcal L$ scheme, a $w^{\rm eob}$ potential scheme, and SEOB-PM—against a common set of equal-mass, nonspinning numerical-relativity scattering simulations Swain, Pratten, and Schmidt 2025, §§ III–IV.

Their results support both positions above. At lower energies, incorporating known high-order information and appropriate resummation can bring the scattering angle close to numerical data. At higher energies, however, the expected PM hierarchy can fail: a nominally higher order can perform worse, and particular completions develop divergences or excessive attraction. Their numerical set reached Lorentz factors beyond earlier comparisons, and Padé resummation of NR-informed radial potentials improved stability in the tested nonspinning equal-mass sector, but the authors explicitly left unequal masses and spin for future study Swain, Pratten, and Schmidt 2025, § V, pp. 18–19, PDF.

This is a concrete limitation, not a verdict that all resummation fails. It shows why “contains 4PM information” and “is more accurate in the strong field” are different claims. It also shows why calibrating a pseudo-higher-order term and testing it on the same scattering set measures fit quality rather than independent prediction.

A controlled comparison protocol

Use the same analytic seed, gauge, and physical parameters for every candidate. First verify symbolic re-expansion through all known PN, PM, and spin orders. Then apply these independent checks:

1. Set $\nu=0$ and recover exact Schwarzschild or Kerr geodesic quantities, not only a weak-field series.
2. Compare a finite-$u$, first-order-in-$\nu$ quantity with self-force data that was not used to choose the form.
3. Compare circular binding energy and periastron advance, which probe different potential combinations.
4. Compare scattering angles across energy and angular momentum, including points near but outside any tuning set.
5. Search for poles, complex Hamiltonians, lost turning points, discontinuities, and nonmonotonic parameter dependence before interpreting a mismatch.

The comparison must separate analytic completion from calibration. If a Padé model also changes fitted coefficients, waveform attachment, and nonquasicircular factors, its final mismatch does not isolate the Padé choice.

Uncertainties, scope, and open questions

As of the cutoff, no theorem selects one global strong-field resummation for generic mass ratio, spin, eccentricity, and scattering energy. Taylor, Padé, logarithmic, factorized, centrifugal-radius, and PM-specific forms each have successful domains and failure modes. Numerical-relativity comparisons are sparse at large mass ratio, near-extremal spin, combined precession and eccentricity, and high-energy near-capture scattering. Self-force data are precise but organized as a mass-ratio expansion. Gauge transformations can also make two apparently different potentials physically equivalent through the known order.

This assessment should be revisited if any of the following occurs:

• new high-accuracy self-force or numerical-relativity data lie materially outside existing calibration domains;
• an independent calculation demonstrates a pole, branch crossing, lost turning point, or discontinuous parameter limit in a standard completion;
• complete higher-PM information reverses the order hierarchy seen in current high-energy scattering tests;
• a same-input, held-out comparison shows one completion consistently improves circular, eccentric, and scattering invariants rather than only a fitted waveform mismatch.

Until such tests are available, a resummed EOB function should be described by its formula, input order, gauge, calibration status, and validated domain—not simply as “the” strong-field potential.

Common pitfalls

Equating re-expansion with validation. Matching every known coefficient proves perturbative consistency only. It says nothing about the unconstrained higher-order terms at strong field.

Ignoring candidate singularities. A Padé pole or logarithmic branch may lie between the initial orbit and physical turning point even when the weak-field series is correct.

Comparing packages instead of choices. Two complete waveform models differ in many sectors. Isolate the potential or mode resummation before assigning cause.

Exercise

Construct the $[1/1]$ Padé approximant to $F(u)=1+c_1u+c_2u^2+O(u^3)$. Locate its pole and state when that pole lies at positive $u$.

Solution

Write

$$P_1^1(u)=\frac{1+a u}{1+b u}.$$

Expanding gives

$$P_1^1 =1+(a-b)u+(b^2-ab)u^2+O(u^3).$$

Matching coefficients yields $a-b=c_1$ and $-bc_1=c_2$, hence

$$b=-\frac{c_2}{c_1}, \qquad a=c_1-\frac{c_2}{c_1},$$

assuming $c_1\neq0$. If also $c_2\neq0$, the denominator vanishes at

$$u_{\rm pole}=-\frac{1}{b}=\frac{c_1}{c_2}.$$

The pole lies at positive $u$ when $c_1$ and $c_2$ have the same sign. If $c_2=0$, the approximant reduces to $1+c_1u$ and has no finite pole. If $c_1=0$ but $c_2\neq0$, a normalized $[1/1]$ approximant cannot match the series at all: $a=b$ forces the quadratic coefficient to vanish. Whether an existing pole is physically harmful depends on whether it falls inside the radial interval used by the model. Rational continuation can therefore introduce a singularity absent from its Taylor seed even while matching every supplied coefficient.

References

• Buonanno, Alessandra, and Thibault Damour. “Effective One-Body Approach to General Relativistic Two-Body Dynamics.” Physical Review D 59, 084006 (1999). DOI. Open PDF.
• Damour, Thibault, Bala R. Iyer, and Alessandro Nagar. “Improved Resummation of Post-Newtonian Multipolar Waveforms from Circularized Compact Binaries.” Physical Review D 79, 064004 (2009). DOI. Open PDF.
• LISA Consortium Waveform Working Group. “Waveform Modelling for the Laser Interferometer Space Antenna.” Living Reviews in Relativity 28, 9 (2025). DOI.
• Pompili, Lorenzo, et al. “Laying the Foundation of the Effective-One-Body Waveform Models SEOBNRv5: Improved Accuracy and Efficiency for Spinning Nonprecessing Binary Black Holes.” Physical Review D 108, 124035 (2023). DOI. Open PDF.
• Swain, Shaun, Geraint Pratten, and Patricia Schmidt. “Strong Field Scattering of Black Holes: Assessing Resummation Strategies.” Physical Review D 111, 064048 (2025). DOI. Open PDF.