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Amplitude Methods for the Binary Problem orIntegrals,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]

Gregor Kälin

QU Day 14.09.2021

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.
• PM-EFT (generic structure at arbitrary loop order): $$\int \dd^Dq\frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^m} \underbrace{\int \dd^D\ell_1\cdots\dd^D\ell_L\frac{\delta(\ell_1\cdot u_{a_1})\cdots\delta(\ell_L\cdot u_{a_L})}{(\ell_1\cdot u_{b_1}\pm i0)^{i_1}\cdots(\ell_L\cdot u_{b_L}\pm i0)^{i_L}(\textrm{sq. props})}}_{\textrm{Cut Feynman integrals with linear and square propagators}}$$ Square propagators: $$[\ell_1^2-i0]$$, $$[(\ell_1-q)^2-i0]$$, ...
• Amplitude approaches: Linear propagators instead of delta functions. Related to above families by (reversed) unitarity: $$-2\pi i\frac{(-1)^{\nu-1}}{(\nu-1)!}\delta^{(\nu-1)}(A)=\frac{1}{(A+i0)^\nu}-\frac{1}{(A-i0)^\nu}$$

Challenges

• Radiation: In-In vs. In-Out or the KMOC setup
$$\langle \Delta p\rangle = \langle\textrm{in}| \Delta p | \textrm{in}\rangle \neq \langle\textrm{in} | \Delta p | \textrm{out} \rangle$$
• In-in boundary conditions: doubling of fields in the WL EFT action.
• Integrand computation becomes significantly harder.
• The gravitons have retarded/advanced propagators. Amplitude methods are optimized for Feynman propagators.
• In the amplitudes approach, the KMOC formalism [1811.10950] solves this problem by writing observables as combinations of amplitudes and cut amplitudes.
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?
• Boundary conditions: So far, in the potential region: IBP + (Schwinger-type) parametrization; these integrals essentially correspond to PN integrals.
• Other regions are harder. Some success with one further region (corresponding to two on-shell graviton legs).
• Going from Feynman to retarded/advanced propagator introduces yet another difficulty.
• The boundary integrals are just numbers (at each order in $$\epsilon$$): Numerics!
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).
• Differential equations and function space: The elliptic sector at three loops is already challenging to solve (canonicalize). What's gonna happen at higher orders?
• Our system contains iterated integrals over elliptic integrals, but they cancel out in the observables.
• In the future? Surviving iterated elliptics? Calabi-Yau?
• Elliptics&friends: Similar challenges as in Amplitudes, precision frontier, Higgs physics, ...
• "Good" set of masters? Transcendental structure, canonical, avoiding spurious cancellations, iterable, PN-correspondence,...?
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?

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).