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]
[2008.06047]

Gregor Kälin

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] $$\chi^\textrm{tot} = \chi^\textrm{local}+\chi^\textrm{non-local}$$

PM-EFT for a worldline action coupled to GR

• Purely classical approach
• Systematic, extension to finite size and spin exists (see Rafael's talk)
• Perturbative expansion in \(G\): can use particle physics/amplitudes toolbox
• Today: only conservative effects in the potential region

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

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

We optimized the EH-Lagrangian by cleverly choosing gauge-fixing terms and adding total derivatives. 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: $$\Delta p^\mu_1= m_1\Delta v^\mu_1 = - \eta^{\mu\nu} \sum_n \int_{-\infty}^{+\infty} \dd\tau_1 {\partial \cL_n \over\partial x^\nu_1}\,,$$ Physical scattering angle is then simply $$2\sin\left(\frac{\chi}{2}\right) = \frac{|\Delta \bp_{1{\rm cm}} |}{p_\infty}= \frac{\sqrt{-\Delta p_1^2}}{p_\infty}$$

Integration

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

• Always one delta function per loop momentum of the form \(\delta(\ell\cdot u_a)\).
• The other combination might appear as a linear propagator in iterations.
• Automatically land in soft classical integrals (see Michael's, Julio's, Gabriele's and Carlo's talk)
• 3PM: a single set of square propagators captures all integrals (i.e. the cut H-family).
• Need a subset of integrals discussed in [BCRSSZ 18; Parra-Martinez, Ruf, Zeng 20]
• In total 876 different integrals (including different \(\pm i0\) prescriptions)
• 4PM: we were able to embed all integrals using two families of square propagators.
• In total 79332 different integrals (including different \(\pm i0\) prescriptions)

IBP relations

[Chetyrkin, Tkackov 81; Gehrmann, Remiddi 2000]

• Very efficient algorithms using integration by part and Lorentz invariance identities to reduce to small subset of independent integrals, e.g. [Laporta 00, Lee 10]
• Implemented in many public packages, e.g. LiteRed, FIRE, Kira, Reduze, AIR.
• Delta functions behave like linear propagators under IBPs (exponent = derivative). Additionally, integrals with negative power delta functions vanish.
• Use CutDS->{...} in LiteRed
• Use RESTRICTIONS={...} in FIRE
• Significant speed-up!
• We use a combination of LiteRed+FIRE6, 3PM: ~5 minutes, 4PM: ~7 hours (4 cores, 32GB RAM)
• 3PM: 7 master integrals without linear propagators, 2 master integrals with linear propagators.
• 4PM: so far we brought the system down to 149 master integrals. (But we know that we are missing some symmetries...)

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].

• This method for our integrals was discussed in [Parra-Martinez, Ruf, Zeng 20].
• Single scale \(\gamma=u_1\cdot u_2=(x^2-1)/(2x)\): \(\partial/\partial x \vec I = \mathbb{M}\cdot \vec I\)
• Having found a canonical form, series exansion in \(\epsilon\) is very simple
• Boundary conditions are 3D (due to delta functions) static integrals, already familiar from PN-EFT.

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 ]

• Simple 5-pt GR Feynman rules ✔
• Integrand ✔
• Map to two basic integral families ✔
• IBP reduction + symmetries ✔ (can we do better?)
• Solve DEQs: different approaches under consideration
• Directly solve the DEQs
• Find canonical form (many packages: epsilon, Fuchsia, Canonica, INITIAL)
• Numerics + reconstruction
• Boundary conditions: in progress
• Masters without linear propagators ✔
• Masters with linear propagators: can be reduced to 2D integrals using symmetrization trick (see e.g. [Cheng, Wu 87; Saotome, Akhoury 13; Parra-Martinez, Ruf, Zeng 20])

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: $$\bp(r,E) = p_\infty^2\left( 1+ \sum_{n=1}^\infty f_n(E) \left(\frac{GM}{r}\right)^n \right) \,,\quad \frac{\chi}{2} = \sum_{n=1}^\infty \chi^{(n)}_b(E) \left(\frac{GM}{b}\right)^n $$ $$\widehat{\chi}_b^{(n)} = \frac{2}{\sqrt{\pi}}\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n+1}{2})}\chi^{(n)}_b\,,\quad g_\sigma^{(n)} = \frac{2(2-n)^{\Sigma^{\ell} - 1}}{\prod_{\ell} (2\sigma^{\ell})!!}$$

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

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?

Conclusions and outlook

• We have a systematic and efficient setup to study the gravitational 2-body problem in its full glory.
• Integration is the bottleneck, but there is a lot of new technology around to help us.
• Can we improve our setup by making contact to the eikonal?
• 4PM is not far!
• Using Firsov, we can resum certain contributions to all orders in G.
• Will also help us for analytic continuation of radiation, radiation-reaction

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).