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A Post-Minkowskian Framework and the Boundary-To-Bound Dictionary for Gravitational Dynamics

Work with Christoph Dlapa, Zhengwen Liu & Rafael Porto
[1910.03008] [1911.09130]
[2006.01184] [2007.04977]
[2008.06047] [2106.XXXX]

scattering orbit

Gregor Kälin

erc desy eu
GGI 19.05.2021



Boundary-to-Bound (B2B) Relations


A worldline action coupled to GR

[Rafael's talk]

Full theory in-out

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

The path to complete \(\Delta p\) / \(\chi\): in-in

EFT action

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

We optimized the EH-Lagrangian by cleverly chosing gauge-fixing terms and adding total derivatives: 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 $$2\sin\left(\frac{\chi}{2}\right) = \frac{|\Delta \bp_{1{\rm cm}} |}{p_\infty} \overset{\text{cons.}}{=} \frac{\sqrt{-\Delta p_1^2}}{p_\infty}$$


Results in the potential region

[2106.XXXX Dlapa, GK, Liu, Porto] [GK, Liu, Porto 20]

$$\frac{\chi}{2} = \sum_{n=1} \chi^{(n)}_b \left(\frac{GM}{b}\right)^n $$

Up to 4PM, agreeing with [Westpfahl, Goller 79][BCRSSZ 18/19] [BPRRSSZ 21]: $$\begin{align} \frac{\chi^{(1)}_b}{\Gamma} &= \frac{2\gamma^2-1}{\gamma^2-1}\\ \frac{\chi^{(2)}_b}{\Gamma} &= \frac{3\pi}{8} \frac{5\gamma^2-1}{\gamma^2-1}\\ \frac{\chi^{(3)}_b}{\Gamma} &= \frac{1}{(\gamma^2-1)^{3/2}}\Bigg[-\frac{4\nu}{3} \gamma\sqrt{\gamma^2-1}(14\gamma^2+25)\frac{(64\gamma^6-120\gamma^4+60\gamma^2-5)(1+2\nu(\gamma-1))}{3(\gamma^2-1)^{3/2}}\\ &\quad- 8\nu (4\gamma^4-12\gamma^2-3)\Arcsinh\sqrt{\frac{\gamma-1}{2}}\,\Bigg] \end{align}$$ with \(\gamma = u_1\cdot u_2\), \(\Gamma=\sqrt{1+2\nu(\gamma-1)}\), \(\nu=m_1m_2/M^2\), \(\gamma=(1+x^2)/(2x)\)

4PM result in the potential region


Completing 4PM: What's next


Generic structure to any loop order (potential + radiation): $$ \int \dd^Dq\frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^m} \underbrace{\int \dd^D\ell_1\cdots\dd^D\ell_L\frac{\delta(\ell_1\cdot u_{a_1})\cdots\delta(\ell_L\cdot u_{a_L})}{(\ell_1\cdot u_{b_1}\pm i0)^{i_1}\cdots(\ell_L\cdot u_{b_L}\pm i0)^{i_L}(\textrm{sq. props})}}_{\textrm{Cut Feynman integrals with linear and square propagators}} $$

Strategy @ 4PM


Some interesting features

Boundary-To-Bound (B2B) Dictionary

[1911.09130, w/ Porto] [1910.03008, w/ Porto]
Conservative motion described by a Hamiltonian: $$H(\bp,r) = E \quad \Longrightarrow \quad \bp(r,E)$$
scattering vs. orbit
Using \(J=p_\infty b\), we have shown that the radii are related by an analytic continuation $$\begin{align*} r_-(J) &= r_{\textrm{min}}(b)\\ r_+(J) &= r_-(-J) = r_{\textrm{min}}(-b) \end{align*}$$ with \(b\in i\mathbb{R}\) for bound kinematics \(p_\infty^2\leq0\).

Angle to orbital Elements: Firsov

[1911.09130, w/ Porto] [1910.03008, w/ Porto]
The scattering angle contains all the information. $$r_{\textrm{min}} \overset{\textrm{Firsov}}{=} b \exp\left[ -\frac{1}{\pi} \int_{b}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-b^2}}\right] =b \prod_{n=1}^\infty e^{-\frac{(GM)^n\chi_b^{(n)}(E)\Gamma\left(\frac{n}{2}\right)}{b^n\sqrt{\pi}\Gamma\left(\frac{n+1}{2}\right)}}$$ with $$\frac{\chi}{2} = \sum_{n=1} \chi^{(n)}_b \left(\frac{GM}{b}\right)^n $$

Angle to periastron advance

[1911.09130, w/ Porto] [1910.03008, w/ Porto] 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
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: \(J=p_\infty b\), \(\bar\bp = \bp/p_\infty\).
\(\textcolor{desyOrange}{\tiny r_\textrm{min}(-J)}\) \(\textcolor{desyOrange}{\tiny r_\textrm{min}(J)}\) $$\begin{align} &=2J \int_{r_\textrm{min}(J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &\quad-2J \int_{r_\textrm{min}(-J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &=\chi(J,E)+\chi(-J,E)+2\pi \end{align}$$

Angle to radial action

[1911.09130, w/ Porto] [1910.03008, w/ Porto]
We can get the radial action from the angle by integrating $$\chi(J,E) = \frac{\dd}{\dd J} \cS_r^\text{hyp}(J,E)$$ The radial action for hyperbolic and elliptic motion are themselves related by a similar analytic continuation $$(G M \mu) i_r^\text{hyp}(J,E) = \cS_r^\text{hyp}(J,E) = \frac{2}{2 \pi}\int_{r_\text{min}(J,E)}^\infty p_r(J,E,r)\dd r$$ $$(G M \mu) i_r^\text{ell}(J,E) = \cS_r^\text{ell}(J,E) = \frac{2}{\pi}\int_{r_-(J,E)}^{r_+(J,E)} p_r(J,E,r)\dd r$$ $$\Rightarrow i_r^\text{ell}(J)=i_r^\text{hyp}(J)- i_r^\text{hyp}(-J)$$ In PM expanded form: $$ i_r^\text{ell}(j,\cE) \equiv \frac{{\cal S}_r}{GM\mu} = {\rm sg}(\hat p_\infty)\chi^{(1)}_j(\cE) - j \left(1 + \frac{2}{\pi} \sum_{n=1} \frac{\chi^{(2n)}_j({\cE})}{(1-2n)j^{2n}}\right) $$

Analytic continuation for aligned spins.

[1911.09130, w/ Porto] Works for aligned spin. 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.

PM to PN: The Magic of Firsov

firsov Let's consider a simple example $${\Delta \Phi}(j,\cE)= \sum_{n=1} \Delta \Phi_j^{(2n)}(\cE)/j^{2n}$$ with \(j=J/(G M \mu)\). We just established $$\Delta \Phi_j^{(2n)}(\cE)= 4\, \chi^{(2n)}_j(\cE)$$ Let's assume we don't know anything about \(\chi^{(4)}_j\). Does this mean we don't know anything about \(\Delta \Phi_j^{(4)}\)? There's lot of lower PN information that our PM results up to \(\chi_j^{(3)}\) should contain. Where are they?

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.

Back to our problem

$$\frac{1}{4}\Delta \Phi_j^{(4)}= \chi^{(4)}_j = \left(\frac{p_\infty}{\mu}\right)^4\chi_b^{(4)}=\left(\frac{p_\infty}{\mu}\right)^4\frac{3\pi}{16}(2f_1f_3+f_2^2+2f_4)$$ and in turn $$\begin{align} f_1&=2\chi_b^{(1)}\\ f_2&=\frac{4}{\pi}\chi_b^{(2)}\\ f_3&=\frac{1}{3}\left(\chi_b^{(1)}\right)^3+\frac{4}{\pi}\chi_b^{(1)}\chi_b^{(2)}+\chi_b^{(3)} \end{align}$$ Prefectly reproduces 1 and 2PN information at order \(j^{-4}\).

Generalization for logs and poles

Pulling out some \((\dots)^\epsilon\) [2106.XXXX Dlapa, GK, Liu, Porto] $$\begin{align} \bp^2(r,E) &= p_\infty^2(E)\left[1+\sum_{n=1}^\infty f_n(E)\left(\frac{G M}{r}\right)^nr^{2n\epsilon}\right]\\ \frac{1}{2}\chi(b,E) &= \sum_{n=1}^\infty \left[\chi_b^{(n)}(E)\left(\frac{GM}{b}\right)^n b^{2n\epsilon}\right] \end{align}$$ allows us easily generalize $$f_n = \sum_{\sigma\in\cP(n)}g_\sigma^{(n)}\prod_\ell \left(\hat{\chi}_b^{(\sigma_\ell)}\right)^{\sigma^\ell}$$ with $$\begin{align} \hat{\chi}_b^{(n)} &= \frac{2}{\sqrt{\pi}}\frac{\Gamma\left(\frac{n(1-2\epsilon)}{2}\right)}{\Gamma\left(\frac{n(1-2\epsilon)+1}{2}\right)}\chi_b^{(n)}\,, & g_\sigma^{(n)} &= \frac{2(2-n(1-2\epsilon))^{\Sigma_\sigma}}{\prod_l(2\sigma^\ell)!!}\end{align}$$

Explicitly $$\begin{align} f_1 &= \frac{2 \Gamma \left(\frac{1}{2}-\epsilon \right)}{\sqrt{\pi } \Gamma (1-\epsilon )}\chi_b^{(1)}\\ f_2 &= \frac{4 \epsilon \Gamma \left(\frac{1}{2}-\epsilon \right)^2}{\pi \Gamma (1-\epsilon )^2}\left(\chi_b^{(1)}\right)^2 +\frac{2 \Gamma (1-2 \epsilon )}{\sqrt{\pi } \Gamma \left(\frac{3}{2}-2 \epsilon \right)}\chi_b^{(2)}\\ f_3 &= \frac{(1-6 \epsilon )^2 \Gamma \left(\frac{1}{2}-\epsilon \right)^3}{3 \pi ^{3/2} \Gamma (1-\epsilon )^3}\left(\chi_b^{(1)}\right)^3 +\frac{2 (6 \epsilon -1) \Gamma (1-2 \epsilon ) \Gamma \left(\frac{1}{2}-\epsilon \right)}{\pi \Gamma \left(\frac{3}{2}-2 \epsilon \right) \Gamma (1-\epsilon )}\chi_b^{(1)} \chi_b^{(2)} +\frac{2 \Gamma \left(\frac{3}{2}-3 \epsilon \right)}{\sqrt{\pi } \Gamma (2-3 \epsilon )}\chi_b^{(3)}\\ f_4 &= \frac{2 (4 \epsilon -1)^3 \Gamma \left(\frac{1}{2}-\epsilon \right)^4}{3 \pi ^2 \Gamma (1-\epsilon )^4}\left(\chi_b^{(1)}\right)^4 +\frac{(2-8 \epsilon )^2 \Gamma (1-2 \epsilon ) \Gamma \left(\frac{1}{2}-\epsilon \right)^2}{\pi ^{3/2} \Gamma \left(\frac{3}{2}-2 \epsilon \right) \Gamma (1-\epsilon )^2}\left(\chi_b^{(1)}\right)^2 \chi_b^{(2)}\\ &\quad+\frac{4 (4 \epsilon -1) \Gamma \left(\frac{3}{2}-3 \epsilon \right) \Gamma \left(\frac{1}{2}-\epsilon \right)}{\pi \Gamma (2-3 \epsilon ) \Gamma (1-\epsilon )}\chi_b^{(1)} \chi_b^{(3)} +\frac{2 \Gamma (2-4 \epsilon )}{\sqrt{\pi } \Gamma \left(\frac{5}{2}-4 \epsilon \right)}\chi_b^{(4)} +\frac{ (8 \epsilon -2) \Gamma (1-2 \epsilon )^2}{\pi \Gamma \left(\frac{3}{2}-2 \epsilon \right)^2}\left(\chi_b^{(2)}\right)^2 \end{align}$$





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).