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(Quantum) Field Theory from the LHC to the Einstein Telescope

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

scattering orbit

Gregor Kälin

kaw desy slac
Quantum Universe Day 17.11.2020
erc eu

Merger breakdown


inspiral: analytic (many cycles!)

PN vs PM.

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

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

merger: numerical relativity (high velocities!)

ringdown: blackhole perturbation theory

Bound-to-boundary (B2B) relations


PM-EFT for a worldline action coupled to GR

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 systematically implement in this formalism [see Zhengwen's talk]

From the EFT action to trajectories

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

E.o.m. from variation of the action 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 $$2\sin\left(\frac{\chi}{2}\right) = \frac{|\Delta \bp_{1{\rm cm}} |}{p_\infty}= \frac{\sqrt{-\Delta p_1^2}}{p_\infty}$$



$$\frac{\chi}{2} = \sum_{n=1} \chi^{(n)}_b \left(\frac{GM}{b}\right)^n $$ 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 \(\gamma = u_1\cdot u_2\), \(\Gamma=\sqrt{1+2\nu(\gamma-1)}\), \(\nu=m_1m_2/M^2\).

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 $${\widetilde{\cM}}(r,E) \equiv \frac{1}{2E}\int \frac{\dd^3\bq}{(2\pi)^3}\, {\cal M}(\bq,\bp^2=p_\infty^2(E)) e^{-i\bq\cdot \br}$$ Relation to scattering angle can be inverted using Firsov's formula [Firsov 53].

Orbital Elements

scattering vs. orbit
$$r_{\textrm{min}} = \tilde r_- \overset{\textrm{Firsov}}{=} b \exp\left[ -\frac{1}{\pi} \int_{b}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-b^2}}\right]$$ 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 could now compute the potential (we derived a formula to all orders in PM expansion)! But there's a more direct way to get observables.
We already saw that the periastron advance can be extracted from the angle/amplitude. In PM language: $$\Delta\Phi = \pi \frac{\widetilde\cM_2}{\mu^2M^2 j^2} +\frac{3\pi}{4}\frac{1}{M^4 \mu^4 j^4}\big(\widetilde\cM_2^2+2\widetilde\cM_1\widetilde\cM_3+2p_\infty^2\widetilde\cM_4\big)+\cdots$$

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




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).

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.


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 \(\cF_2 \equiv f_2/f_1^2\)

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 \(x \equiv (GM \Omega_{\rm circ})^{2/3}\).