Post-Minkowskian EFT meets conservative potential at NNNLO

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?