## Boundary-To-Bound (B2B) relations

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]

$$\Delta p_1 =-\eta^{\mu\nu}\sum_n\int\dd\tau\frac{\partial\cL_n}{\partial x_1^\nu}$$ $$2\sin\left(\frac{\chi}{2}\right)=\frac{|\Delta \bp_{1\textrm{cm}}|}{p_\infty}$$

*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: $$-2\pi i\frac{(-1)^{\nu-1}}{(\nu-1)!}\delta^{(\nu-1)}(A)=\frac{1}{(A+i0)^\nu}-\frac{1}{(A-i0)^\nu}$$

*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.

*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!

*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,...?

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).