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$$ \definecolor{desyOrange}{RGB}{242,142,0} \DeclareMathOperator{\Arcsinh}{arcsinh} \DeclareMathOperator{\Arctan}{arctan} \DeclareMathOperator{\arcosh}{arcosh} \newcommand\dd{{\mathrm d}} \newcommand{\bb}{{\mathbf b}} \newcommand{\bk}{{\mathbf k}} \newcommand{\bl}{{\mathbf l}} \newcommand{\bm}{{\mathbf m}} \newcommand{\bn}{{\mathbf n}} \newcommand{\bp}{{\mathbf p}} \newcommand{\bq}{{\mathbf q}} \newcommand{\br}{{\mathbf r}} \newcommand{\bw}{{\mathbf w}} \newcommand{\bx}{{\mathbf x}} \newcommand{\by}{{\mathbf y}} \newcommand{\bz}{{\mathbf z}} \newcommand\cM{\mathcal{M}} \newcommand\cO{\mathcal{O}} \newcommand{\cD}{\mathcal{D}} \newcommand\cE{\mathcal{E}} \newcommand\cS{\mathcal{S}} \newcommand\cP{\mathcal{P}} \newcommand\cF{\mathcal{F}} \newcommand\cL{\mathcal{L}} \newcommand\Mp{M_{\rm Pl}} $$

Post-Minkowskian EFT meets conservative potential at NNNLO

Continuation of work with R. Porto & Z. Liu in
[1910.03008] [1911.09130]
[2006.01184] [2007.04977]

scattering orbit

Gregor Kälin

kaw desy slac
QCD meets Gravity VI
erc eu

Scattering angle: state of the art

$$\frac{\chi}{2} = \sum_{n=1} \chi^{(n)}_b \left(\frac{GM}{b}\right)^n $$ 3PM result: QCD meets Gravity 2018 [BCRSSZ 18; Cheung, Solon 20; GK, Liu, Porto 20] $$\begin{align} \frac{\chi^{(1)}_b}{\Gamma} &= \frac{2\gamma^2-1}{\gamma^2-1}\\ \frac{\chi^{(2)}_b}{\Gamma} &= \frac{3\pi}{8} \frac{5\gamma^2-1}{\gamma^2-1}\\ \frac{\chi^{(3)}_b}{\Gamma} &= \frac{1}{(\gamma^2-1)^{3/2}}\Bigg[-\frac{4\nu}{3} \gamma\sqrt{\gamma^2-1}(14\gamma^2+25)\frac{(64\gamma^6-120\gamma^4+60\gamma^2-5)\Gamma^2}{3(\gamma^2-1)^{3/2}}\\ &\quad- 8\nu (4\gamma^4-12\gamma^2-3)\Arcsinh\sqrt{\frac{\gamma-1}{2}}\,\Bigg] \end{align}$$ with \(\gamma = u_1\cdot u_2 [=\sigma]\), \(\Gamma=\sqrt{1+2\nu(\gamma-1)}[=h(\gamma,\nu)]\), \(\nu=m_1m_2/M^2\).
Directly feeds into the radial action (\(J=p_\infty b=G M \mu j\)): $$i_r(j,\cE) \equiv \frac{{\cal S}_r}{GM\mu} = {\rm sg}(\hat p_\infty )\chi^{(1)}_j(\cE) - j \left(1 + \frac{2}{\pi} \sum_{n=1} \frac{\chi^{(2n)}_j({\cE})}{(1-2n)j^{2n}}\right)$$

What do we know about \(\chi^{(4)}_b\)?

Schwarzschild limit (\(\nu=0\)): $$\chi^{(4),\textrm{Sch}}_b=\frac{105 \pi \left(33 \gamma ^4-18 \gamma ^2+1\right)}{128 \left(\gamma ^2-1\right)^2}$$ Lots of PN data [Bini, Damour, Geralico 20; + Laporta, Mastrolia 20] $$\chi^\textrm{tot} = \chi^\textrm{local}+\chi^\textrm{non-local}$$
chiLocal chiNonLocal

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

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 choosing gauge-fixing terms and adding total derivatives.
Without field redefinitions:
  • 2-point Lagrangian: 2 terms
  • 3-point Lagrangian: 6 terms
  • 4-point Lagrangian: 18 terms (\(\leftarrow\) 3PM)
  • 5-point Lagrangian: 36 terms (\(\leftarrow\) 4PM)
With field redefinitions:
  • 2-point Lagrangian: 2 terms
  • 3-point Lagrangian: 4 terms
  • 4-point Lagrangian: 12 terms
We chose to not use field redefinitions to preserve the simple one-particle coupling to the WL. (Maybe we should for finite-size effects + spin?)

PM deflection

Having integrated out potential gravitons we have: $$S_{\rm eff} = \sum_{n=0}^\infty \int \dd\tau_1\,G^n \cL_n [x_1(\tau_1),x_2(\tau_2)]$$ 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: $$\Delta p^\mu_1= m_1\Delta v^\mu_1 = - \eta^{\mu\nu} \sum_n \int_{-\infty}^{+\infty} \dd\tau_1 {\partial \cL_n \over\partial x^\nu_1}\,,$$
At 3PM:
  • \(\cL_1\left[b_a+u_a\tau_a+\delta^{(1)}x_a+\delta^{(2)}x_a\right]\)
  • \(\cL_2\left[b_a+u_a\tau_a+\delta^{(1)}x_a\right]\)
  • \(\cL_3\left[b_a+u_a\tau_a \right]\)
At 4PM:
  • \(\cL_1\left[b_a+u_a\tau_a+\delta^{(1)}x_a+\delta^{(2)}x_a+\delta^{(3)}x_a\right]\)
  • \(\cL_2\left[b_a+u_a\tau_a+\delta^{(1)}x_a+\delta^{(2)}x_a\right]\)
  • \(\cL_3\left[b_a+u_a\tau_a+\delta^{(1)}x_a \right]\)
  • \(\cL_4\left[b_a+u_a\tau_a\right]\)
Physical scattering angle is then simply $$2\sin\left(\frac{\chi}{2}\right) = \frac{|\Delta \bp_{1{\rm cm}} |}{p_\infty}= \frac{\sqrt{-\Delta p_1^2}}{p_\infty}$$



For an alternative derivation of the same integrand using a dynamical WL see Gustav's talk.
Generic structure: $$ \int \dd^Dq\frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^m} \underbrace{\int \dd^D\ell_1\cdots\dd^D\ell_L\frac{\delta(\ell_1\cdot u_{a_1})\cdots\delta(\ell_L\cdot u_{a_L})}{(\ell_1\cdot u_{b_1}\pm i0)^{i_1}\cdots(\ell_L\cdot u_{b_L}\pm i0)^{i_L}(\textrm{sq. props})}}_{\textrm{Cut Feynman integrals with linear and square propagators}} $$

IBP relations

[Chetyrkin, Tkackov 81; Gehrmann, Remiddi 2000]

DEQs and the canonical form @ 3PM

Compute the master integrals using differential equations and their canonical form [Kotikov, Remiddi, Gehrmann 91, 98, 99; Henn 13, 14]. Final Fourier transform is known to all orders: $$\Delta p_a^\mu = \int \dd^Dq (A \overbrace{q^\mu}^{-i \frac{\partial}{\partial b_\mu}} + \underbrace{B u_1^\mu + C u_2^\mu}_{\substack{\textrm{bootstrap from}\\ \textrm{on-shell condition}\\ (\Delta p_a+p_a)^2=p_a^2}}) \frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^{m+n}}$$ (NB: using on-shell constraints we only need terms \(\sim \partial/\partial b^\mu\). Eikonal?)

Where are we at NNNLO?

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

Analytic continuation and Firsov's formula

Let us get a feeling of the analytic continuation and some funny games we can play! $$i_r(j,\cE) \equiv \frac{{\cal S}_r}{GM\mu} = {\rm sg}(\hat p_\infty )\chi^{(1)}_j(\cE) - j \left(1 + \frac{2}{\pi} \sum_{n=1} \frac{\chi^{(2n)}_j({\cE})}{(1-2n)j^{2n}}\right)$$ What about \(\chi_j^{(3)}\)?

Firsov's formula

Let's do a detour to the c.o.m. momentum along the trajectory: $$H(r,\bp) = E \Rightarrow \bp(r,E)$$ Relation to angle most easily extracted from Firsov's formula $$\bar\bp^2(r,E) = \exp\left[ \frac{2}{\pi} \int_{r|\bar\bp(r,E)|}^\infty \frac{\chi_b(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-r^2\bar\bp^2(r,E)}}\right]$$ In PM language: $$\bp(r,E) = p_\infty^2\left( 1+ \sum_{n=1}^\infty f_n(E) \left(\frac{GM}{r}\right)^n \right) \,,\quad \frac{\chi}{2} = \sum_{n=1}^\infty \chi^{(n)}_b(E) \left(\frac{GM}{b}\right)^n $$
$$f_n = \sum_{\sigma\in\mathcal{P}(n)}g_\sigma^{(n)} \prod_{\ell} \left(\widehat{\chi}_b^{(\sigma_{\ell})}\right)^{\sigma^{\ell}}$$
$$\widehat{\chi}_b^{(n)} = \frac{2}{\sqrt{\pi}}\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n+1}{2})}\chi^{(n)}_b\,,\quad g_\sigma^{(n)} = \frac{2(2-n)^{\Sigma^{\ell} - 1}}{\prod_{\ell} (2\sigma^{\ell})!!}$$

Radial action at 4PM

$$i_r (j,\cE) = -j + \frac{\hat p_\infty^2}{ \sqrt{-\hat p_\infty^2}} \frac{f_1}{2} + \frac{\hat p_\infty^2}{2j} f_2 + \frac{\hat p_\infty^4}{8j^3}\Big( f_2^2 +2 f_1f_3+2f_4 \Big) +\cdots$$ where in turn $$\begin{align} f_1 &= \frac{2\chi_j^{(1)}}{\hat p_\infty}\,,\\ f_2 &= \frac{4 \chi_j^{(2)}}{\pi \hat p_\infty^2}\,,\\ f_3 &= \frac{\left(\chi_j^{(1)}\right)^3}{3 \hat p_\infty^3} - \frac{4 \chi_j^{(1)} \chi_j^{(2)}}{\pi \hat p_\infty^3} + \frac{\chi_j^{(3)}}{\hat p_\infty^3}\,,\\ f_4 &= \frac{8 \chi_j^{(4)}}{3 \pi \hat p_\infty^4}+\dots \quad (\leftarrow\textrm{That's our missing guy}) \end{align}$$


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

Conclusions and outlook

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).