(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]
[2008.06047]

Gregor Kälin

Merger breakdown

inspiral: analytic (many cycles!)

• Goal: Compute effective two-body Hamiltonian (+ radiative losses) $$\begin{aligned} H\sim G(1+v^2+v^4+v^6+\dots&)\\ + G^2(1+v^2+v^4+\dots&)\\ + G^3(1+v^2+\dots&) + \dots\\ \end{aligned} $$
• Various directions:
  • Conservative potential
  • Spinning objects
  • Radiation
  • 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 \(G\)
  • "Classical GR" methods [Bertotti; Westpfahl, Goller; Bel, Damour, Deruelle, Ibanez, Martin; Ledvinka, Schäfer, Bicak; ...]
  • Using perturbation theory: Scattering amplitudes [Cheung, Rothstein, Solon; Neill, Rothstein; Bjerrum-Bohr, Damgaard, Festuccia, Planté, Vanhove; Guevara, Ochirov, Vines; Damour; Cristofoli, Di Vecchia, Heissenberg, Veneziano, Russo; Parra-Martinez, Ruf, Zeng;... ]
  • First 3PM result [Bern, Cheung, Roiban, Shen, Solon, Zeng 2018/2019]

merger: numerical relativity (high velocities!)

• Perturbation theory breaks down.
• See breakthrough papers: [Pretorius '05; Campanelli et al. '06, Baker et al. '06]

ringdown: blackhole perturbation theory

• Model the system as excitations of a single massive body.

Bound-to-boundary (B2B) relations

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

From the EFT action to trajectories

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

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 $$2\sin\left(\frac{\chi}{2}\right) = \frac{|\Delta \bp_{1{\rm cm}} |}{p_\infty}= \frac{\sqrt{-\Delta p_1^2}}{p_\infty}$$

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 \(\gamma = u_1\cdot u_2\), \(\Gamma=\sqrt{1+2\nu(\gamma-1)}\), \(\nu=m_1m_2/M^2\).

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: 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

$$r_{\textrm{min}} = \tilde r_- \overset{\textrm{Firsov}}{=} b \exp\left[ -\frac{1}{\pi} \int_{b}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-b^2}}\right]$$ 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

• 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]

Conclusions

• Theorists need to prepare for next generation gravitational wave detectors: More templates!
• We developed an 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

• Analytic continuation for arbitrary spin? Radiation? Radiation-reaction? Tail?
• PM-EFT with spin: soon!
• Conservative scattering angle at 4PM: We have all the technology.
• We can resum certain quantities. More work required. (Ask me about it if you are interested!)
• Non-perturbative nature of identities invites for numerical studies.

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!

• 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:

• 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?

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