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# A Post-Minkowskian EFT for the Gravitational Two-Body Problem

Work with Rafael Porto & Zhengwen Liu in
[1910.03008] [1911.09130]
[2006.01184] [2007.04977]
[2008.06047]

Gregor Kälin

AEI 27.01.2021

## Motivation: Merger breakdown

### merger: numerical relativity (high velocities!)

• Expensive simulations, not many cycles
• Breakthrough 2005 [Pretorius '05; Campanelli et al. '06, Baker et al. '06]

### inspiral: analytic (many cycles!)

• Goal: Describe the dynamics of two massive objects in GR analytically
• Various directions:
• Conservative dynamics
• Spinning objects
• Radiation reaction (tail effects)
• Inner structure & tidal deformations (BH vs NS)
• ...
• Post-Newtonian: expansion in $$v^2\sim\frac{Gm}{r}\ll 1$$
• [Droste, Lorentz 1917; ...; Einstein, Infeld, Hoffmann 1938]
• [Blanchet, Damour, Iyer, Faye, Bernard, Bohe’, Buonanno, Marsat; Jaranowski, Schäfer, Steinhoff; Will, Wiseman; Flanagan, Hinderer, Vines; Goldberger, Porto, Rothstein; Kol, Levi, Smolkin; Cristofoli, Mastrolia, Foffa, Sturani, Torres-Bobadilla; Blümlein, Maier, Marquard, Schäfer;...]
• Post-Minkowskian: expansion in $$G$$
• Small mass-ratio/gravitational self-force: $$m_2/m_1\ll 1$$
• [Regge, Wheeler 56; Zerilli 70; Teukolsky 72]
• [Fujita, Poisson, Sasaki, Shibata, Khanna, Hughes, Bernuzzi, Harms, Nagar, Deitweiler, Whiting, Mino, Quinn, Wald, Tanaka, Barack, Ori, Pound, van de Meent, ...]

# PN vs PM.

State of the art for the conservative potential (non-spinning) Buonanno@Amplitudes2018

0PN 1PN 2PN 3PN 4PN 5PN 6PN 7PN
0PM $$~~1~~$$ $$v^2$$ $$v^4$$ $$v^6$$ $$v^8$$ $$v^{10}$$ $$v^{12}$$ $$v^{14}$$ $$v^{16}$$
1PM $$1/r$$ $$v^2/r$$ $$v^4/r$$ $$v^6/r$$ $$v^8/r$$ $$v^{10}/r$$ $$v^{12}/r$$ $$v^{14}/r$$
2PM $$1/r^2$$ $$v^2/r^2$$ $$v^4/r^2$$ $$v^6/r^2$$ $$v^8/r^2$$ $$v^{10}/r^2$$ $$v^{12}/r^2$$
3PM $$1/r^3$$ $$v^2/r^3$$ $$v^4/r^3$$ $$v^6/r^3$$ $$v^8/r^3$$ $$v^{10}/r^3$$
4PM $$1/r^4$$ $$v^2/r^4$$ $$v^4/r^4$$ $$v^6/r^4$$ $$v^8/r^4$$
5PM $$1/r^5$$ $$v^2/r^5$$ $$v^4/r^5$$ $$v^6/r^5$$

## PM-EFT for a worldline action coupled to GR

• We are only interested in classical physics
• Perturbative expansion in $$G$$: can use particle physics/amplitudes toolbox
• Today: only interested in conservative effects

### Full theory

\begin{align} S_{\rm EH} &= -2\Mp^2 \int \dd^4x \sqrt{-g} \, R[g]\\ S_{\rm pp} &= - \sum_a m_a \int \dd\sigma_a \sqrt{g_{\mu\nu}(x^\alpha_{a}(\sigma)) v_{a}^\mu(\sigma_a) v_{a}^\nu (\sigma_a)} + \dots\\ &\rightarrow -\sum_a \frac{m_a}{2} \int \dd\tau_a\, g_{\mu\nu}(x_{a}(\tau_a)) v_{a}^\mu(\tau_a) v_{a}^\nu (\tau_a)+\dots \end{align}

Extensions to finite-size effects and spinning bodies are simple to implement in this formalism.

### EFT action

$$e^{i S_{\rm eff}[x_a] } = \int \cD h_{\mu\nu} \, e^{i S_{\rm EH}[h] + i S_{\rm GF}[h] + i S_{\rm pp}[x_a,h]}$$

We optimized the EH-Lagrangian by cleverly chosing gauge-fixing terms and adding total derivatives:
• 2-point Lagrangian: 2 terms
• 3-point Lagrangian: 6 terms
• 4-point Lagrangian: 18 terms
• 5-point Lagrangian: 36 terms
These numbers could even be further reduced by field redefinitions, but we don't want higher point WL-couplings.

### PM deflection

In a Post-Minkowskian expansion: $$S_{\rm eff} = \sum_{n=0}^\infty \int \dd\tau_1\,G^n \cL_n [x_1(\tau_1),x_2(\tau_2)]$$ with $$\cL_0 = - \frac{m_1}{2} \eta_{\mu\nu} v_1^\mu(\tau_1) v_1^\nu(\tau_1)$$ E.o.m. from variation of the action $$-\eta^{\mu\nu}\frac{\dd}{\dd\tau_1} \left(\frac{\partial \cL_0}{\partial v^\nu_1}\right) = m_1 \frac{\dd v_1^\mu}{\dd\tau_1} = -\eta^{\mu\nu}\left(\sum_{n=1}^{\infty} \frac{\partial \cL_n}{\partial x^\nu_1(\tau_1)} - \frac{\dd}{\dd\tau_1} \left(\frac{\partial \cL_n}{\partial v^\nu_1}\right)\right)$$ allows us to compute the trajectories order by order: $$x^\mu_a(\tau_1) = b^\mu_a + u^\mu_a \tau_a + \sum_n G^n \delta^{(n)} x^\mu_a (\tau_a)$$ with $$b=b_1-b_2$$ the impact parameter and $$u_a$$ the incoming velocty at infinity, fulfilling $$u_1\cdot u_2 = \gamma\,, \quad u_a\cdot b = 0\,.$$

### Scattering angle.

First we compute the deflection using above trajectories: $$\Delta p^\mu_1= m_1\Delta v^\mu_1 = - \eta^{\mu\nu} \sum_n \int_{-\infty}^{+\infty} \dd\tau_1 {\partial \cL_n \over\partial x^\nu_1}\,,$$ Physical scattering angle is then simply $$2\sin\left(\frac{\chi}{2}\right) = \frac{|\Delta \bp_{1{\rm cm}} |}{p_\infty}= \frac{\sqrt{-\Delta p_1^2}}{p_\infty}$$

### Integration

Generic structure: $$\int \dd^Dq\frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^m} \underbrace{\int \dd^D\ell_1\cdots\dd^D\ell_L\frac{\delta(\ell_1\cdot u_{a_1})\cdots\delta(\ell_L\cdot u_{a_L})}{(\ell_1\cdot u_{b_1}\pm i0)^{i_1}\cdots(\ell_L\cdot u_{b_L}\pm i0)^{i_L}(\textrm{sq. props})}}_{\textrm{Cut Feynman integrals with linear and square propagators}}$$
• Integration by part identities (IBPs): reduction to masters [Chetyrkin, Tkachov '81,...]
• Differential equations in $$x$$: $$\gamma= (x^2+ 1)/(2x)$$ [Parra-Martinez, Ruf, Zeng '20]
• Canonical basis: solve order by order in $$\epsilon$$ [Henn 13' 14']
• Static boundary conditions: potential region sufficient for conservative part $$\rightarrow$$ already known from PN-EFT
Final Fourier transform is known to all orders: $$\Delta p_a^\mu = \int \dd^Dq (A \overbrace{q^\mu}^{-i \frac{\partial}{\partial b_\mu}} + \underbrace{B u_1^\mu + C u_2^\mu}_{\substack{\textrm{bootstrap from}\\ \textrm{on-shell condition}\\ (\Delta p_a+p_a)^2=p_a^2}}) \frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^{m+n}}$$

### Results.

$$\frac{\chi}{2} = \sum_{n=1} \chi^{(n)}_b \left(\frac{GM}{b}\right)^n$$ Up to 3PM (agreeing with [Bern, Cheung, Roiban, Shen, Solon, Zeng 2018/2019]): \begin{align} \frac{\chi^{(1)}_b}{\Gamma} &= \frac{2\gamma^2-1}{\gamma^2-1}\\ \frac{\chi^{(2)}_b}{\Gamma} &= \frac{3\pi}{8} \frac{5\gamma^2-1}{\gamma^2-1}\\ \frac{\chi^{(3)}_b}{\Gamma} &= \frac{1}{(\gamma^2-1)^{3/2}}\Bigg[-\frac{4\nu}{3} \gamma\sqrt{\gamma^2-1}(14\gamma^2+25)\frac{(64\gamma^6-120\gamma^4+60\gamma^2-5)(1+2\nu(\gamma-1))}{3(\gamma^2-1)^{3/2}}\\ &\quad- 8\nu (4\gamma^4-12\gamma^2-3)\Arcsinh\sqrt{\frac{\gamma-1}{2}}\,\Bigg] \end{align} with $$\gamma = u_1\cdot u_2$$, $$\Gamma=\sqrt{1+2\nu(\gamma-1)}$$, $$\nu=m_1m_2/M^2$$.

### News @4PM

Update from last week: potential contributions @4PM presented by [Bern, Parra-Martinez, Roiban, Ruf, Shen, Solon, Zeng '21]

• New elliptic/hypergeometric functions
• Spurious IR divergence & problematic high-energy limit due to split into potential & radiaton

### Where are we at 4PM?

[work with C. Dlapa, Z. Liu, G. Mogull, R. Porto ]

• Simple 5-pt GR Feynman rules ✔
• Integrand ✔
• Map to two basic integral families ✔
• IBP reduction + symmetries ✔: 89 masters
• Solve DEQs: different approaches under consideration
• Directly solve the DEQs
• Find canonical form: one family solved ✔
• Numerics + reconstruction
• Boundary conditions: in progress
• Masters without linear propagators ✔
• Masters with linear propagators: can be reduced to 2D integrals using symmetrization trick (see e.g. [Cheng, Wu 87; Saotome, Akhoury 13; Parra-Martinez, Ruf, Zeng 20])
• Trying to get full soft region!

## Boundary-To-Bound (B2B) dictionary

Conservative motion described by a Hamiltonian: $$H(\bp,r) = E \quad \Longrightarrow \quad \bp(r,E)$$ Scattering angle: $$\chi(b,E) +\pi = 2b \int_{r_{\rm min}}^\infty \frac{\dd r}{r\sqrt{r^2\bar\bp^2(r,E)-b^2}}$$ with $$r_\textrm{min}$$ the positive real root of $$p_r$$: $$p_r^2(r,E)=\bp^2(r,E)-J^2/r^2$$.
Periastron advance: $$\Delta \Phi + 2\pi = 2J \int_{r_-}^{r_+} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}$$ Don't forget: $$J=p_\infty b$$, $$\bar\bp = \bp/p_\infty$$.

## Interlude: Impetus formula

In [1910.03008, w/ R. Porto] we found a formula relating the classical IR-finite part of the 2-to-2 scattering amplitude to the momentum along the trajectory:
$$\mathbf{p}^2(r,E) = p_\infty^2(E) + \widetilde{\cM}(r,E)$$
with $${\widetilde{\cM}}(r,E) \equiv \frac{1}{2E}\int \frac{\dd^3\bq}{(2\pi)^3}\, {\cal M}(\bq,\bp^2=p_\infty^2(E)) e^{-i\bq\cdot \br}$$ or in PM expanded form: $$\widetilde{\cal M}_n(E) = P_n(E)$$ with $$\widetilde{\cal M}(r,E) = \sum_{n=1}^\infty \widetilde{\cal M}_n(E) \left(\frac{G}{r}\right)^n,\quad\bp^2(r,E) = p_\infty^2(E) +\sum_{n=1}^\infty P_n(E) \left(\frac{G}{r}\right)^n$$

### Firsov's formula

Let us invert $$\chi(b,E) = -\pi + 2b \int_{r_{\rm min}}^\infty \frac{\dd r}{r\sqrt{r^2\bar\bp^2(r,E)-b^2}} = \sum_{n=1} \chi^{(n)}_b(E) \left(\frac{GM}{b}\right)^n$$ [Firsov '53]: dependence on $$r_\textrm{min}$$ drops out
$$\bar\bp^2(r,E) = \exp\left[ \frac{2}{\pi} \int_{r|\bar{\bp}(r,E)|}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-r^2\bar\bp^2(r,E)}}\right] = 1 + \sum_{n=1}^\infty f_n(E) \left(\frac{GM}{r}\right)^n$$
These integrals are easy to perform in a PM-expanded form and one finds: $$\chi_b^{(n)} = \frac{\sqrt{\pi}}{2} \Gamma\left(\frac{n+1}{2}\right)\sum_{\sigma\in\mathcal{P}(n)}\frac{1}{\Gamma\left(1+\frac{n}{2} -\Sigma^\ell\right)}\prod_{\ell} \frac{f_{\sigma_{\ell}}^{\sigma^{\ell}}}{\sigma^{\ell}!}$$ The inversion thereof also exists.

### Orbital Elements

vs.
$$r_{\textrm{min}} = \tilde r_- \overset{\textrm{Firsov}}{=} b \exp\left[ -\frac{1}{\pi} \int_{b}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-b^2}}\right] =b \prod_{n=1}^\infty e^{-\frac{(GM)^n\chi_b^{(n)}(E)\Gamma\left(\frac{n}{2}\right)}{b^n\sqrt{\pi}\Gamma\left(\frac{n+1}{2}\right)}}$$ Do an analytic continuation in $$J=p_\infty b$$ with $$b\in i\mathbb{R}$$ ($$p_\infty^2\leq0$$): \begin{align*} r_-(J) &= r_{\textrm{min}}(b)\\ r_+(J) &= r_-(-J) = r_{\textrm{min}}(-b) \end{align*} These are the two real positive roots of $$p_r$$: $$p_r^2(r,E)=\bp^2(r,E)-J^2/r^2$$ with $$J=p_\infty b$$
$$\textcolor{desyOrange}{\tiny r_\textrm{min}(-J)}$$ $$\textcolor{desyOrange}{\tiny r_\textrm{min}(J)}$$ \begin{align} &=2J \int_{r_\textrm{min}(J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &\quad-2J \int_{r_\textrm{min}(-J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &=\chi(J,E)+\chi(-J,E)+2\pi \end{align}

## Observables for the bound state

We just saw that the periastron advance can be extracted from the angle: $$\Delta\Phi(J,E) = \chi(J,E)+\chi(-J,E)$$ It constains almost all information to compute the full radial action!

### The radial action

$$\frac{\Delta \Phi+2\pi}{2\pi} = - \frac{\partial \cS_r(J,\cE)}{\partial J}= -\frac{\partial}{\partial J} \frac{1}{\pi} \int_{r_-}^{r_+} \sqrt{\bp^2(r,{\cal E})-J^2/r^2}$$ Simply integrate in $$J$$ and find the boundary term: $$i_r = \frac{{\cal S}_r}{GM\mu} = {\rm sg}(\hat p_\infty)\chi^{(1)}_j(\cE) - j \left(1 + \frac{2}{\pi} \sum_{n=1} \frac{\chi^{(2n)}_j({\cE})}{(1-2n)j^{2n}}\right)$$ with $$\chi_j^{(n)} = \left(\frac{p_\infty}{M\nu}\right)^n \chi_b^{(n)}$$. Or in amplitudes language: $$i_r = -j + \frac{{\widetilde \cM}_1}{2 \sqrt{ -\hat p^2_\infty}M \mu^2} + \frac{{\widetilde \cM}_2}{2 jM^2\mu^2} + \frac{{\widetilde \cM}_2^2 +2 {\widetilde \cM}_1{\widetilde \cM}_3+2 p_\infty^2 {\widetilde \cM}_4}{8j^3M^4\mu^4}+\cdots$$

### Observables: examples

• Periastron advance: $$\frac{\Delta\Phi}{2\pi} = -\frac{\partial}{\partial j} \cS_r(j,\cE)-1$$
• Periastron-to-periastron period: $$\frac{T_p}{2\pi} = \frac{1}{\mu}\frac{\partial}{\partial \cE} \cS_r(j,\cE)$$
• Radial frequency: $$\Omega_r (j,\cE) = \frac{2\pi}{T_p}$$
• Periastron frequency: $$\Omega_p (j,\cE) = \frac{\Delta\Phi}{T_p}$$
• Azimuthal frequency: $$\Omega_\phi\equiv\Omega_r+\Omega_p=\frac{2\pi}{T_p}\left(1+\frac{\Delta\Phi}{2\pi}\right)$$
• Averaged redshift: $$\langle z_a\rangle = 1+\frac{\partial \mu}{\partial m_a} \cE -\Omega_r \frac{\partial}{\partial m_a}\cS_r(j,\cE,m_a)$$ [LeTiec 15]

### Classical potential

We can also extract the PM-expanded conservative two-body potential. We iteratively solve $$H(r,\bp^2) = E = \sqrt{\bp^2+m_1^2}+\sqrt{\bp^2+m_2^2} + V(r,\bp^2)$$ using $$E = E_1 + E_2 = \sqrt{p_\infty^2+m_1^2}+\sqrt{p_\infty^2+m_2^2}$$ and $$p_\infty^2(E) = \bp^2(r,E) - \sum_i P_i(E) \left(\frac{G}{r}\right)^i\,,$$ for the PM coefficient of the potential $$V(r,\bp^2) = \sum_{i=0}^{\infty} \frac{c_i(\bp^2)}{i!} \left(\frac{G}{r}\right)^i\,.$$

We find for the leading coefficient: $$c_0(\bp^2) = E(\bp^2) = E_1(\bp^2)+E_2(\bp^2) \equiv \sqrt{\bp^2+m_1^2} + \sqrt{\bp^2+m_2^2}$$ And a complicated (but purely combinatorial) expression for higher orders: $$c_i({\bp}^2) = \sum_{k=1}^{k=i} \frac{\sqrt{\pi}}{2\Gamma\left(\frac{3}{2}-k\right)} \frac{E_1({\bp}^2)^{2k-1}+E_2({\bp}^2)^{2k-1}}{(E_1({\bp}^2)E_2({\bp}^2))^{2k-1}}B_{i,k}\left({\cal G}_1({\bp}^2),\dots, {\cal G}_{i-k+1}({\bp}^2)\right)\,,$$ with Bell polynomials $$B_{i,k}$$ and $${\cal G}_m({\bp}^2) = -\sum_{s=0}^m\sum_{\ell=0}^s\frac{m!}{s!}P_{m-s}^{(\ell)}(c_0({\bp}^2))B_{s,\ell}\left(c_1({\bp}^2),\dots,c_{s-\ell+1}({\bp}^2)\right)$$

## PM-EFT: Tidal effects for extended objects

$$S_{\rm pp} \mathrel{+}= \sum_a \!\int \!\dd\tau_a\!\left(c^{(a)}_{E^2} E_{\mu\nu} E^{\mu\nu} +c^{(a)}_{B^2} B_{\mu\nu} B^{\mu\nu} - c^{(a)}_{{\tilde E}^2} E_{\mu\nu\alpha} E^{\mu\nu\alpha} - c^{(a)}_{{\tilde B}^2} B_{\mu\nu\alpha} B^{\mu\nu\alpha}+\cdots \right)$$ with operators \begin{align} E_{\alpha\beta} &= R_{\mu\alpha\nu\beta} v^\mu v^\nu\,, & B_{\alpha\beta} &= R^\star_{\mu\alpha\nu\beta} v^\mu v^\nu, \\ E_{\alpha\beta\gamma} &= \nabla^\perp_{\{\alpha} R_{\beta\rho\gamma \} \nu} v^\rho v^\nu\,, & B_{\alpha\beta\gamma} &= \nabla^\perp_{\{\alpha} R^\star_{\beta\rho\gamma \} \nu} v^\rho v^\nu\end{align} Use same setup as before and compute:

• Reproduced quadrupole corrections at NLO PM [Cheung, Solon 20]
• Novel octupole corrections at NLO PM
• Analytic continuation works as before
• Binding energy for circular orbits at $$\cO(G^3)$$

## PM-EFT: Spinning extended objects

[Porto; Porto, Rothstein; ...]
$$S_{\rm pp} \mathrel{+}= -\frac{1}{2}\sum_a \int \dd\tau_a S_{ab}(\tau_a)\omega_\mu^{ab}(\tau_a)\dot{x}_a^\mu(\tau_a)\,,$$ with spin connection $$\omega_\mu^{ab}$$.
• We can use the same setup for the PM expansion
• Need to simultaneously solve e.o.m. for the spin tensor: $$\frac{\dd S^{ab}}{\dd\tau} = \{S^{ab},\cL_\textrm{int}\}$$
• The amplitude at 2PM for spin-orbit effects has been worked out by [Bern, Luna, Roiban, Shen, Zeng 20]

## Analytic continuation for aligned spins.

Idea: extend our map to the aligned spins for binary BH problem.
Motion is still in a plane!
$$\frac{\chi(J,\cE)+\chi(-J,\cE)}{2\pi} = \frac{\Delta\Phi(J,\cE)}{2\pi}$$

where $$J$$ is now the total the total angular momentum, i.e. orbital angular momentum + spins.

• Explicit checks for known PN and PM results work neatly!
• Relies on the invariance of the (canonical) radial momentum $$p_r$$ under $$J\rightarrow -J$$, which is true for a quasi-isotropic gauge (given to us automatically by the amplitudes construction).
• We propose a version of the impetus formula to also hold for the aligned spin case.

## Resummation

Let us truncate our theory at given order $$n$$, i.e. $$\cM_m=f_m=0$$ for $$m \geq n$$.
We can try to resum contributions to all orders in $$G$$, e.g. for the scattering angle:

\begin{align*} \frac{\chi[f_1]}{2} &= \Arctan(y/2)\\ \frac{\chi[f_{1,2}]+\pi}{2} &= \frac{1}{\sqrt{1-{\cF}_2 y^2}}\left(\frac{\pi}{2} + \Arctan\left(\frac{y}{2\sqrt{1-{\cF}_2 y^2}}\right)\right) \end{align*}

with $$y \equiv G M f_1/b$$ and $$\cF_2 \equiv f_2/f_1^2$$

• Resummation of $$\Delta\phi$$ works similar to $$\chi$$.
• We can resum parts of $$\cS_r$$.
• We can resum $$f_{1,2}$$ contributions for $$r_\textrm{min}$$ and $$r_\pm$$. ("closed form" for real positive roots of arbitrary order polynomial?)
• Difficult for $$f_{1,2,3}$$. Anyone can do it?

## Conclusions

• We developed a systematic & efficient PM EFT framework, allows us to include tidal effects and spin.
• Mapping (conservative & non-spinning) scattering data to orbital observables does not require a Hamiltonian. Analytic continuation does the job.
• Analytic continuation also works the special case of aligned spins.
• The (classical) amplitude, the scattering angle, and the momentum along the trajectory contain the same information.
• We have these relations in exact form, useful for e.g. numerical computation, as well as in PM expanded form to all orders (purely combinatorial).

## Outlook

• Non-perturbative map invites for numerical studies: Self-force limit?
• Analytic continuation for arbitrary spin? Radiation? Radiation-reaction? Tail?
• Full angle at 4PM: We are getting there!
• Input from self-force computation (numerical & analytical)?
• We can resum certain quantities. More work required.
• New type of integrals: What's the space of functions to all loop orders?
• Compute integrals numerically (and analytically reconstruct?): Machine learning?

This research is supported by the ERC-CoG “Precision Gravity: From the LHC to LISA” provided by the European Research Council (ERC) under the European Union’s H2020 research and innovation programme (grant No. 817791), by the DFG under Germany’s Excellence Strategy ‘Quantum Universe’ (No. 390833306), and by the Knut and Alice Wallenberg Foundation (grant KAW 2018.0441).

## Circular orbit

Solve $$r_+(J) = r_-(J)$$ $$\Leftrightarrow -2 \sum_{n=0}^\infty \left(\frac{1}{\sqrt{\pi}}\left(\frac{GM}{b}\right)^{2n+1}\frac{\Gamma\left(\frac{2n+1}{2}\right)}{\Gamma(n+1)}\chi_b^{(2n+1)}\right) = i \pi + 2\pi i \mathbb{N}$$ to find $$j(\cE)$$ and compute the radial frequency: $$GM\Omega_{\rm circ} = \left(\frac{\dd\, j({\cal E})}{\dd\, {\cal E}}\right)^{-1}$$ Need to resum for a truncated theory $$f_i=0$$ for $$i\geq n$$!
We can invert to write the binding energy $$\epsilon\equiv -2 \cE$$: \begin{aligned} \epsilon =&x \left[1 - \frac{x}{12}(9+\nu) - \frac{x^2}{8}\left(27 -19\nu + \frac{\nu^2}{3}\right)\right.+\frac{x^3}{32}\left(\frac{535}{6}-\frac{5585\nu}{6}+135\nu^2-\frac{35\nu^3}{162}\right) \\ &+ \left.\frac{x^4}{384}\left(-10171+\frac{559993}{15}\nu-\frac{34027\nu^2}{3}+\frac{11354\nu^3}{9}+\frac{77\nu^4}{81}\right) + {\cal O}(x^5)\right] \end{aligned} using the standard PN parameter $$x \equiv (GM \Omega_{\rm circ})^{2/3}$$.