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A Post-Minkowskian Framework and the Boundary-To-Bound Dictionary for Gravitational Dynamics

Work with Christoph Dlapa, Zhengwen Liu & Rafael Porto
[1910.03008] [1911.09130]
[2006.01184] [2007.04977]
[2008.06047] [2106.XXXX]

Gregor Kälin

GGI 19.05.2021

A worldline action coupled to GR

[Rafael's talk]
• Purely classical
• We will use EFT methods: Full action $$\rightarrow$$ effective action $$\rightarrow$$ deflection/flux/waveform/...
• Perturbative expansion in $$G$$: can use particle physics/amplitudes toolbox
• Can easily include: finite size, spin, $$n$$-body

Full theory in-out

\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}

The path to complete $$\Delta p$$ / $$\chi$$: in-in

• To get correct boundary conditions ($$i0$$ prescription for graviton propagators) we need to use the in-in formalism (doubling of fields) \begin{align} \cS[h_1,h_2] = \cS_\textrm{EH}[h_1] - \cS_\textrm{EH}[h_2] -\sum_{A=1}^2 \frac{\kappa m_A}{2}\int\dd\tau_A &\left[h_{1,\mu\nu}(x_{1,A}(\tau_A))\dot{x}_{1,A}^\mu(\tau_A)\dot{x}_{1,A}^\nu(\tau_A)\right.\\ &\quad\left.-h_{2,\mu\nu}(x_{2,A}(\tau_A))\dot{x}_{2,A}^\mu(\tau_A)\dot{x}_{2,A}^\nu(\tau_A)\right] \end{align}
• Not as bad as it sounds, Keldysh variables clean up Feynman rules.
• For 4PM conservative tail of RR the in-in formalism is not needed: In-out integrand with "self-energy" diagrams is sufficient (only two graviton lines go on-shell). We have this integrand.

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 chosing gauge-fixing terms and adding total derivatives:
• 2-point Lagrangian: 2 terms
• 3-point Lagrangian: 6 terms
• 4-point Lagrangian: 18 terms
• 5-point Lagrangian: 36 terms
These numbers could even be further reduced by field redefinitions, but we don't want higher point WL-couplings.

PM deflection

In a Post-Minkowskian expansion: $$S_{\rm eff} = \sum_{n=0}^\infty \int \dd\tau_1\,G^n \cL_n [x_1(\tau_1),x_2(\tau_2)]$$ 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 $$2\sin\left(\frac{\chi}{2}\right) = \frac{|\Delta \bp_{1{\rm cm}} |}{p_\infty} \overset{\text{cons.}}{=} \frac{\sqrt{-\Delta p_1^2}}{p_\infty}$$

Results in the potential region

[2106.XXXX Dlapa, GK, Liu, Porto] [GK, Liu, Porto 20]

$$\frac{\chi}{2} = \sum_{n=1} \chi^{(n)}_b \left(\frac{GM}{b}\right)^n$$

Up to 4PM, agreeing with [Westpfahl, Goller 79][BCRSSZ 18/19] [BPRRSSZ 21]: \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)(1+2\nu(\gamma-1))}{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 $$\gamma = u_1\cdot u_2$$, $$\Gamma=\sqrt{1+2\nu(\gamma-1)}$$, $$\nu=m_1m_2/M^2$$, $$\gamma=(1+x^2)/(2x)$$

Completing 4PM: What's next

• For the conservative RR tail we have the integrand. We do not expect new integrals. And even if, then not elliptic $$=$$ easy! (no $$\pi^3$$ in PN tail). We are putting it together now!
• In-in integrands are straightforwardly obtained. Automated code will give us full 4PM integrand soon.
• Strategies to get the soft boundary conditions have been worked out @3PM [Di Vecchia, Heissenberg, Russo, Veneziano 21; Herrmann, Parra-Martinez, Ruf, Zeng 21][Mao's, Carlo's and Michael's talk]. We expect that they also apply @4PM.
• In-in: We are gonna get retarded/advanced propagators: Breaks $$\ell\rightarrow-\ell$$ symmetry of propagators which is helpful when relating integrals (i.e. we will get more master integrals).
• $$u_a^\mu$$ contributions cannot be bootstrapped anymore due to dissipative effects: More integrals.

Integration

Generic structure to any loop order (potential + radiation): $$\int \dd^Dq\frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^m} \underbrace{\int \dd^D\ell_1\cdots\dd^D\ell_L\frac{\delta(\ell_1\cdot u_{a_1})\cdots\delta(\ell_L\cdot u_{a_L})}{(\ell_1\cdot u_{b_1}\pm i0)^{i_1}\cdots(\ell_L\cdot u_{b_L}\pm i0)^{i_L}(\textrm{sq. props})}}_{\textrm{Cut Feynman integrals with linear and square propagators}}$$
• Immediately land on soft, classical, single scale integrals. No expansion needed.
• Use modern amplitudes methods [Mao's + Michael's talk]
• One delta function per loop $$\rightarrow$$ half of the linear WL propagators are cut
• @3PM: One single family + $$i0^+$$ prescription for linear propagators. We have solved the DEQs for all $$\sim20$$ master integrals (including those appearing for dissipative and spin effects).
• @4PM: Only two families of square propagators + $$i0^+$$ prescription for linear propagators.

Some interesting features

• We solve all integrals to needed order in $$\epsilon$$, all orders in $$v$$. No resummation!
• Precanonical form is sufficient for blocks containing elliptic integrals. The $$\cO(\epsilon^0)$$ DEQ can be solved for diagonal elements: 3x3 elliptic blocks $$\rightarrow$$ 3rd order differential equation which is solved by a quadratic expression in terms of complete elliptic integrals. Off-diagonal integrals can then be solved order by order in $$\epsilon$$.
• This induces iterated elliptic integrals, e.g. $$\int_0^x \dd x\frac{8 \left(x^2 K\left(1-x^2\right)^2-x^2-E\left(1-x^2\right)^2+1\right)}{x \left(x^2-1\right)}=-8 \left(K\left(1-x^2\right) E\left(1-x^2\right)+\log \left(\frac{x}{4}\right)\right)$$ Fortunately, all that were needed can be evaluated in terms of elliptic integrals.
• Because of unphysical $$1/\epsilon$$ pole we need to expand the integrand up to $$\cO(\epsilon)$$ to match BPRRSSZ
• Overkill! Should not be necessary. Tail contributions will kill the pole and such unphysical contributions.
• Schwarzschild contributions are known to all order. Could be used to constrain boundary conditions.

Boundary-To-Bound (B2B) Dictionary

[1911.09130, w/ Porto] [1910.03008, w/ Porto]
Conservative motion described by a Hamiltonian: $$H(\bp,r) = E \quad \Longrightarrow \quad \bp(r,E)$$
vs.
• Scattering: $$r_\textrm{min}=\tilde{r}_-$$ is the positive real root of $$p_r$$: $$p_r^2(r,E)=\bp^2(r,E)-(p_\infty b)^2/r^2$$ for positive binding energy.
• Bound: $$r_\pm$$ are positive real roots of $$p_r$$: $$p_r^2(r,E)=\bp^2(r,E)-J^2/r^2$$ for negative binding energy.
Using $$J=p_\infty b$$, we have shown that the radii are related by an analytic continuation \begin{align*} r_-(J) &= r_{\textrm{min}}(b)\\ r_+(J) &= r_-(-J) = r_{\textrm{min}}(-b) \end{align*} with $$b\in i\mathbb{R}$$ for bound kinematics $$p_\infty^2\leq0$$.

Angle to orbital Elements: Firsov

[1911.09130, w/ Porto] [1910.03008, w/ Porto]
The scattering angle contains all the information. $$r_{\textrm{min}} \overset{\textrm{Firsov}}{=} b \exp\left[ -\frac{1}{\pi} \int_{b}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-b^2}}\right] =b \prod_{n=1}^\infty e^{-\frac{(GM)^n\chi_b^{(n)}(E)\Gamma\left(\frac{n}{2}\right)}{b^n\sqrt{\pi}\Gamma\left(\frac{n+1}{2}\right)}}$$ with $$\frac{\chi}{2} = \sum_{n=1} \chi^{(n)}_b \left(\frac{GM}{b}\right)^n$$

[1911.09130, w/ Porto] [1910.03008, w/ Porto] Scattering angle: $$\chi(b,E) +\pi = 2b \int_{r_{\rm min}}^\infty \frac{\dd r}{r\sqrt{r^2\bar\bp^2(r,E)-b^2}}$$ with
Periastron advance: $$\Delta \Phi + 2\pi = 2J \int_{r_-}^{r_+} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}$$ Don't forget: $$J=p_\infty b$$, $$\bar\bp = \bp/p_\infty$$.
$$\textcolor{desyOrange}{\tiny r_\textrm{min}(-J)}$$ $$\textcolor{desyOrange}{\tiny r_\textrm{min}(J)}$$ \begin{align} &=2J \int_{r_\textrm{min}(J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &\quad-2J \int_{r_\textrm{min}(-J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &=\chi(J,E)+\chi(-J,E)+2\pi \end{align}

[1911.09130, w/ Porto] [1910.03008, w/ Porto]
We can get the radial action from the angle by integrating $$\chi(J,E) = \frac{\dd}{\dd J} \cS_r^\text{hyp}(J,E)$$ The radial action for hyperbolic and elliptic motion are themselves related by a similar analytic continuation $$(G M \mu) i_r^\text{hyp}(J,E) = \cS_r^\text{hyp}(J,E) = \frac{2}{2 \pi}\int_{r_\text{min}(J,E)}^\infty p_r(J,E,r)\dd r$$ $$(G M \mu) i_r^\text{ell}(J,E) = \cS_r^\text{ell}(J,E) = \frac{2}{\pi}\int_{r_-(J,E)}^{r_+(J,E)} p_r(J,E,r)\dd r$$ $$\Rightarrow i_r^\text{ell}(J)=i_r^\text{hyp}(J)- i_r^\text{hyp}(-J)$$ In PM expanded form: $$i_r^\text{ell}(j,\cE) \equiv \frac{{\cal S}_r}{GM\mu} = {\rm sg}(\hat p_\infty)\chi^{(1)}_j(\cE) - j \left(1 + \frac{2}{\pi} \sum_{n=1} \frac{\chi^{(2n)}_j({\cE})}{(1-2n)j^{2n}}\right)$$

Analytic continuation for aligned spins.

[1911.09130, w/ Porto] Works for aligned spin. Motion is still in a plane!
$$\frac{\chi(J,\cE)+\chi(-J,\cE)}{2\pi} = \frac{\Delta\Phi(J,\cE)}{2\pi}$$

where $$J$$ is now the total the total angular momentum, i.e. orbital angular momentum + spins.

• Explicit checks for known PN and PM results work neatly!
• Relies on the invariance of the (canonical) radial momentum $$p_r$$ under $$J\rightarrow -J$$, which is true for a quasi-isotropic gauge (given to us automatically by the amplitudes construction).

PM to PN: The Magic of Firsov

Let's consider a simple example $${\Delta \Phi}(j,\cE)= \sum_{n=1} \Delta \Phi_j^{(2n)}(\cE)/j^{2n}$$ with $$j=J/(G M \mu)$$. We just established $$\Delta \Phi_j^{(2n)}(\cE)= 4\, \chi^{(2n)}_j(\cE)$$ Let's assume we don't know anything about $$\chi^{(4)}_j$$. Does this mean we don't know anything about $$\Delta \Phi_j^{(4)}$$? There's lot of lower PN information that our PM results up to $$\chi_j^{(3)}$$ should contain. Where are they?

Firsov's formula

Let us invert $$\chi(b,E) = -\pi + 2b \int_{r_{\rm min}}^\infty \frac{\dd r}{r\sqrt{r^2\bar\bp^2(r,E)-b^2}} = \sum_{n=1} \chi^{(n)}_b(E) \left(\frac{GM}{b}\right)^n$$ [Firsov '53]: dependence on $$r_\textrm{min}$$ drops out
$$\bar\bp^2(r,E) = \exp\left[ \frac{2}{\pi} \int_{r|\bar{\bp}(r,E)|}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-r^2\bar\bp^2(r,E)}}\right] = 1 + \sum_{n=1}^\infty f_n(E) \left(\frac{GM}{r}\right)^n$$
These integrals are easy to perform in a PM-expanded form and one finds: $$\chi_b^{(n)} = \frac{\sqrt{\pi}}{2} \Gamma\left(\frac{n+1}{2}\right)\sum_{\sigma\in\mathcal{P}(n)}\frac{1}{\Gamma\left(1+\frac{n}{2} -\Sigma^\ell\right)}\prod_{\ell} \frac{f_{\sigma_{\ell}}^{\sigma^{\ell}}}{\sigma^{\ell}!}$$ The inversion thereof also exists.

Back to our problem

$$\frac{1}{4}\Delta \Phi_j^{(4)}= \chi^{(4)}_j = \left(\frac{p_\infty}{\mu}\right)^4\chi_b^{(4)}=\left(\frac{p_\infty}{\mu}\right)^4\frac{3\pi}{16}(2f_1f_3+f_2^2+2f_4)$$ and in turn \begin{align} f_1&=2\chi_b^{(1)}\\ f_2&=\frac{4}{\pi}\chi_b^{(2)}\\ f_3&=\frac{1}{3}\left(\chi_b^{(1)}\right)^3+\frac{4}{\pi}\chi_b^{(1)}\chi_b^{(2)}+\chi_b^{(3)} \end{align} Prefectly reproduces 1 and 2PN information at order $$j^{-4}$$.

Generalization for logs and poles

Pulling out some $$(\dots)^\epsilon$$ [2106.XXXX Dlapa, GK, Liu, Porto] \begin{align} \bp^2(r,E) &= p_\infty^2(E)\left[1+\sum_{n=1}^\infty f_n(E)\left(\frac{G M}{r}\right)^nr^{2n\epsilon}\right]\\ \frac{1}{2}\chi(b,E) &= \sum_{n=1}^\infty \left[\chi_b^{(n)}(E)\left(\frac{GM}{b}\right)^n b^{2n\epsilon}\right] \end{align} allows us easily generalize $$f_n = \sum_{\sigma\in\cP(n)}g_\sigma^{(n)}\prod_\ell \left(\hat{\chi}_b^{(\sigma_\ell)}\right)^{\sigma^\ell}$$ with \begin{align} \hat{\chi}_b^{(n)} &= \frac{2}{\sqrt{\pi}}\frac{\Gamma\left(\frac{n(1-2\epsilon)}{2}\right)}{\Gamma\left(\frac{n(1-2\epsilon)+1}{2}\right)}\chi_b^{(n)}\,, & g_\sigma^{(n)} &= \frac{2(2-n(1-2\epsilon))^{\Sigma_\sigma}}{\prod_l(2\sigma^\ell)!!}\end{align}

Explicitly \begin{align} f_1 &= \frac{2 \Gamma \left(\frac{1}{2}-\epsilon \right)}{\sqrt{\pi } \Gamma (1-\epsilon )}\chi_b^{(1)}\\ f_2 &= \frac{4 \epsilon \Gamma \left(\frac{1}{2}-\epsilon \right)^2}{\pi \Gamma (1-\epsilon )^2}\left(\chi_b^{(1)}\right)^2 +\frac{2 \Gamma (1-2 \epsilon )}{\sqrt{\pi } \Gamma \left(\frac{3}{2}-2 \epsilon \right)}\chi_b^{(2)}\\ f_3 &= \frac{(1-6 \epsilon )^2 \Gamma \left(\frac{1}{2}-\epsilon \right)^3}{3 \pi ^{3/2} \Gamma (1-\epsilon )^3}\left(\chi_b^{(1)}\right)^3 +\frac{2 (6 \epsilon -1) \Gamma (1-2 \epsilon ) \Gamma \left(\frac{1}{2}-\epsilon \right)}{\pi \Gamma \left(\frac{3}{2}-2 \epsilon \right) \Gamma (1-\epsilon )}\chi_b^{(1)} \chi_b^{(2)} +\frac{2 \Gamma \left(\frac{3}{2}-3 \epsilon \right)}{\sqrt{\pi } \Gamma (2-3 \epsilon )}\chi_b^{(3)}\\ f_4 &= \frac{2 (4 \epsilon -1)^3 \Gamma \left(\frac{1}{2}-\epsilon \right)^4}{3 \pi ^2 \Gamma (1-\epsilon )^4}\left(\chi_b^{(1)}\right)^4 +\frac{(2-8 \epsilon )^2 \Gamma (1-2 \epsilon ) \Gamma \left(\frac{1}{2}-\epsilon \right)^2}{\pi ^{3/2} \Gamma \left(\frac{3}{2}-2 \epsilon \right) \Gamma (1-\epsilon )^2}\left(\chi_b^{(1)}\right)^2 \chi_b^{(2)}\\ &\quad+\frac{4 (4 \epsilon -1) \Gamma \left(\frac{3}{2}-3 \epsilon \right) \Gamma \left(\frac{1}{2}-\epsilon \right)}{\pi \Gamma (2-3 \epsilon ) \Gamma (1-\epsilon )}\chi_b^{(1)} \chi_b^{(3)} +\frac{2 \Gamma (2-4 \epsilon )}{\sqrt{\pi } \Gamma \left(\frac{5}{2}-4 \epsilon \right)}\chi_b^{(4)} +\frac{ (8 \epsilon -2) \Gamma (1-2 \epsilon )^2}{\pi \Gamma \left(\frac{3}{2}-2 \epsilon \right)^2}\left(\chi_b^{(2)}\right)^2 \end{align}

Hamiltonian

• The effective potential for the two-body Hamiltonian is simple to obtain from the scattering angle/momentum along the trajectory.
• Our formula for the $$c_n(\bp^2)$$ coefficient for the potential in [1910.03008, w/ Porto] still holds for log/poles.
• We explicitly checked the $$c_4(\bp^2)$$ coefficient in BPRRSSZ obtained from the scattering angle with the above formulae.

Conclusions

• PMEFT+B2B: systematic and efficient framework to study the classical gravitational 2-body dynamics.
• Modern integration techniques can handle all the integrals we have found. No resummation needed.
• We have obtained the potential contributions @4PM. More to come soon.
• We are creating artificial problems by splitting into potential+radiation. Computationally there are reasons for this split, but we need to be careful to not overcomplicate our lives. It also complicates comparisons among different approaches and limits.
• Firsov's formula is bridging $$\chi\leftrightarrow \bp^2 \leftrightarrow \cH$$ and $$\text{PN}\leftrightarrow\text{PM}$$.

Outlook

• We have not reached any tool's maximal capacity. 4PM can be performed on a desktop computer.
• First priority: complete 4PM: Tail, dissipative effects.
• Iterated elliptic integrals need to be understood better.
• Study B2B for arbitrary spin and dissipative effects.

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).