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

Amplitude Methods for the Binary Problem
or
Integrals, Integrals, Integrals

Based on work with
Christoph Dlapa, Ryusuke Jinno, Zhengwen Liu, Rafael Porto & Henrique Rubira
[1910.03008] [1911.09130]
[2006.01184] [2007.04977]
[2008.06047] [2106.08276]

scattering orbit

Gregor Kälin

erc desy eu
QU Day 14.09.2021

The binary problem and gravitational waves


waveform

Boundary-To-Bound (B2B) relations

scattering

Overview of different approaches (potential)

PM EFT: \(\Delta p\)

$$\begin{align} \cS &= \cS_\textrm{EH}[h] + \cS_\textrm{WL}[h,x_a]\\ &\Downarrow(\textrm{Feynman diagrams})\\ e^{i\cS_\textrm{eff}} &= \int \cD h e^{i\cS_\textrm{EH}+i\cS_\textrm{GF}+i\cS_\textrm{WL}} \end{align}$$ Perturbative expansion: $$\cS_\textrm{eff}=\sum_{n=0}^\infty\int\dd\tau G^n \cL_n[x_1,x_2]$$ E.o.m. \(\Rightarrow\) deflection:
$$\Delta p_1 =-\eta^{\mu\nu}\sum_n\int\dd\tau\frac{\partial\cL_n}{\partial x_1^\nu}$$ $$2\sin\left(\frac{\chi}{2}\right)=\frac{|\Delta \bp_{1\textrm{cm}}|}{p_\infty}$$

Amplitude

$$\cS = \cS_\textrm{EH}[h] + \cS_\textrm{scalar}[h, \phi_a]$$ Tree level amplitudes: double copy [BCJ '08 '10] $$(\textrm{gravity})=(\textrm{gauge thy})^2$$ Loop level amplitudes: generalized unitarity

EFT matching

$$\cM \overset{!}{=} \cM_\textrm{eff}(V_\textrm{eff})$$ $$\cH(r,p) = \cH_\textrm{kin}(r,p)\!+\! V_\textrm{eff}(r,p)$$ $$\chi=-\pi+2J\int_{r_\textrm{min}}^\infty \frac{\dd r}{r^2\sqrt{p_r(r)^2}}$$ with $$E=H(r^2,p^2=p_r^2+J^2/r^2)$$

Eikonal exp

$$\tilde\cM(b_\textrm{e})\propto\int \dd^{2-2\epsilon}q e^{ib_\textrm{e}\cdot q}\cM(q) $$ Eikonal phase \(\delta(s,b_\textrm{e})\): $$1+i\tilde\cM = e^{2i\delta}(1+2i\Delta)$$ \(\Delta(s,b_\textrm{e})\) quantum remainder $$\sin\left(\frac{\chi}{2}\right)=\frac{\partial}{\partial |b_\textrm{e}|}\frac{\Re(2\delta)}{2 p_\infty}$$

Integration

All approaches lead to similar (scalar) Feynman integrals.

Integration procedure @4PM (potential)

integration

Result @4PM

res4PM

Challenges

With the help of a Master student, Jakob Neef, we are translating our code to C++ (based on GiNaC) to handle the integrand construction more efficiently.
We either need cut amplitudes (on-shell integrations) or retarded/advanced propagators. What is easier?
PN data contains all information for \(v\rightarrow 0\), can we use this for other regions?
Systematic approach at higher loop needed. Boundary IBP relations for other regions? Iterative approach?
We are working on various ideas related to a numerical bootstrap of such integrals (also using machine learning techniques to speed up the numerical algorithms).
A "good" basis could resolve many of our problems, but how to find it? Does it exist?
Where can our friends from related fields help us out?

main

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).