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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]

scattering orbit

Gregor Kälin

erc desy eu
GGI 19.05.2021

Boundary-to-Bound (B2B) Relations

scattering

A worldline action coupled to GR

[Rafael's talk]
  • Purely classical
  • We will use EFT methods: Full action effective action 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

SEH=2MPl2d4xgR[g]Spp=amadσagμν(xaα(σ))vaμ(σa)vaν(σa)+ama2dτagμν(xa(τa))vaμ(τa)vaν(τa)+

The path to complete Δp / χ: in-in

  • To get correct boundary conditions (i0 prescription for graviton propagators) we need to use the in-in formalism (doubling of fields)
    S[h1,h2]=SEH[h1]SEH[h2]A=12κmA2dτA[h1,μν(x1,A(τA))x˙1,Aμ(τA)x˙1,Aν(τA)h2,μν(x2,A(τA))x˙2,Aμ(τA)x˙2,Aν(τA)]
  • 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

eiSeff[xa]=DhμνeiSEH[h]+iSGF[h]+iSpp[xa,h]
graphsEFT

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:
Seff=n=0dτ1GnLn[x1(τ1),x2(τ2)]
with
L0=m12ημνv1μ(τ1)v1ν(τ1)
E.o.m. from variation of the action
ημνddτ1(L0v1ν)=m1dv1μdτ1=ημν(n=1Lnx1ν(τ1)ddτ1(Lnv1ν))
allows us to compute the trajectories order by order:
xaμ(τ1)=baμ+uaμτa+nGnδ(n)xaμ(τa)
with b=b1b2 the impact parameter and ua the incoming velocty at infinity, fulfilling
u1u2=γ,uab=0.

Scattering angle.

First we compute the deflection using above trajectories:
Δp1μ=m1Δv1μ=ημνn+dτ1Lnx1ν,
Physical scattering angle is then simply
2sin(χ2)=|Δp1cm|p=cons.Δp12p

scattering

Results in the potential region

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

χ2=n=1χb(n)(GMb)n

Up to 4PM, agreeing with [Westpfahl, Goller 79][BCRSSZ 18/19] [BPRRSSZ 21]:
χb(1)Γ=2γ21γ21χb(2)Γ=3π85γ21γ21χb(3)Γ=1(γ21)3/2[4ν3γγ21(14γ2+25)(64γ6120γ4+60γ25)(1+2ν(γ1))3(γ21)3/28ν(4γ412γ23)arcsinhγ12]
with γ=u1u2, Γ=1+2ν(γ1), ν=m1m2/M2, γ=(1+x2)/(2x)

4PM result in the potential region

res4PM

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 π3 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 symmetry of propagators which is helpful when relating integrals (i.e. we will get more master integrals).
  • uaμ contributions cannot be bootstrapped anymore due to dissipative effects: More integrals.

Integration

Generic structure to any loop order (potential + radiation):
dDqδ(qu1)δ(qu2)eibq(q2)mdD1dDLδ(1ua1)δ(LuaL)(1ub1±i0)i1(LubL±i0)iL(sq. props)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 half of the linear WL propagators are cut
  • @3PM: One single family + i0+ prescription for linear propagators. We have solved the DEQs for all 20 master integrals (including those appearing for dissipative and spin effects).
  • @4PM: Only two families of square propagators + i0+ prescription for linear propagators.

Strategy @ 4PM

integration

Some interesting features

  • We solve all integrals to needed order in ϵ, all orders in v. No resummation!
  • Precanonical form is sufficient for blocks containing elliptic integrals. The O(ϵ0) DEQ can be solved for diagonal elements: 3x3 elliptic blocks 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 ϵ.
  • This induces iterated elliptic integrals, e.g.
    0xdx8(x2K(1x2)2x2E(1x2)2+1)x(x21)=8(K(1x2)E(1x2)+log(x4))
    Fortunately, all that were needed can be evaluated in terms of elliptic integrals.
  • Because of unphysical 1/ϵ pole we need to expand the integrand up to O(ϵ) 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(p,r)=Ep(r,E)
scattering vs. orbit
  • Scattering: rmin=r~ is the positive real root of pr: pr2(r,E)=p2(r,E)(pb)2/r2 for positive binding energy.
  • Bound: r± are positive real roots of pr: pr2(r,E)=p2(r,E)J2/r2 for negative binding energy.
Using J=pb, we have shown that the radii are related by an analytic continuation
r(J)=rmin(b)r+(J)=r(J)=rmin(b)
with biR for bound kinematics p20.

Angle to orbital Elements: Firsov

[1911.09130, w/ Porto] [1910.03008, w/ Porto]
The scattering angle contains all the information.
rmin=Firsovbexp[1πbχ(b~,E)db~b~2b2]=bn=1e(GM)nχb(n)(E)Γ(n2)bnπΓ(n+12)
with
χ2=n=1χb(n)(GMb)n

Angle to periastron advance

[1911.09130, w/ Porto] [1910.03008, w/ Porto] Scattering angle:
χ(b,E)+π=2brmindrrr2p¯2(r,E)b2
with
Periastron advance:
ΔΦ+2π=2Jrr+drrr2p2(r,E)J2
Don't forget: J=pb, p¯=p/p.
rmin(J) rmin(J)
=2Jrmin(J)drrr2p2(r,E)J22Jrmin(J)drrr2p2(r,E)J2=χ(J,E)+χ(J,E)+2π

Angle to radial action

[1911.09130, w/ Porto] [1910.03008, w/ Porto]
We can get the radial action from the angle by integrating
χ(J,E)=ddJSrhyp(J,E)
The radial action for hyperbolic and elliptic motion are themselves related by a similar analytic continuation
(GMμ)irhyp(J,E)=Srhyp(J,E)=22πrmin(J,E)pr(J,E,r)dr
(GMμ)irell(J,E)=Srell(J,E)=2πr(J,E)r+(J,E)pr(J,E,r)dr
irell(J)=irhyp(J)irhyp(J)
In PM expanded form:
irell(j,E)SrGMμ=sg(p^)χj(1)(E)j(1+2πn=1χj(2n)(E)(12n)j2n)

Analytic continuation for aligned spins.

[1911.09130, w/ Porto] Works for aligned spin. Motion is still in a plane!
χ(J,E)+χ(J,E)2π=ΔΦ(J,E)2π

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 pr under JJ, which is true for a quasi-isotropic gauge (given to us automatically by the amplitudes construction).

PM to PN: The Magic of Firsov

firsov Let's consider a simple example
ΔΦ(j,E)=n=1ΔΦj(2n)(E)/j2n
with j=J/(GMμ). We just established
ΔΦj(2n)(E)=4χj(2n)(E)
Let's assume we don't know anything about χj(4). Does this mean we don't know anything about ΔΦj(4)? There's lot of lower PN information that our PM results up to χj(3) should contain. Where are they?

Firsov's formula

Let us invert
χ(b,E)=π+2brmindrrr2p¯2(r,E)b2=n=1χb(n)(E)(GMb)n
[Firsov '53]: dependence on rmin drops out
p¯2(r,E)=exp[2πr|p¯(r,E)|χ(b~,E)db~b~2r2p¯2(r,E)]=1+n=1fn(E)(GMr)n
These integrals are easy to perform in a PM-expanded form and one finds:
χb(n)=π2Γ(n+12)σP(n)1Γ(1+n2Σ)fσσσ!
The inversion thereof also exists.

Back to our problem

14ΔΦj(4)=χj(4)=(pμ)4χb(4)=(pμ)43π16(2f1f3+f22+2f4)
and in turn
f1=2χb(1)f2=4πχb(2)f3=13(χb(1))3+4πχb(1)χb(2)+χb(3)
Prefectly reproduces 1 and 2PN information at order j4.

Generalization for logs and poles

Pulling out some ()ϵ [2106.XXXX Dlapa, GK, Liu, Porto]
p2(r,E)=p2(E)[1+n=1fn(E)(GMr)nr2nϵ]12χ(b,E)=n=1[χb(n)(E)(GMb)nb2nϵ]
allows us easily generalize
fn=σP(n)gσ(n)(χ^b(σ))σ
with
χ^b(n)=2πΓ(n(12ϵ)2)Γ(n(12ϵ)+12)χb(n),gσ(n)=2(2n(12ϵ))Σσl(2σ)!!

Explicitly
f1=2Γ(12ϵ)πΓ(1ϵ)χb(1)f2=4ϵΓ(12ϵ)2πΓ(1ϵ)2(χb(1))2+2Γ(12ϵ)πΓ(322ϵ)χb(2)f3=(16ϵ)2Γ(12ϵ)33π3/2Γ(1ϵ)3(χb(1))3+2(6ϵ1)Γ(12ϵ)Γ(12ϵ)πΓ(322ϵ)Γ(1ϵ)χb(1)χb(2)+2Γ(323ϵ)πΓ(23ϵ)χb(3)f4=2(4ϵ1)3Γ(12ϵ)43π2Γ(1ϵ)4(χb(1))4+(28ϵ)2Γ(12ϵ)Γ(12ϵ)2π3/2Γ(322ϵ)Γ(1ϵ)2(χb(1))2χb(2)+4(4ϵ1)Γ(323ϵ)Γ(12ϵ)πΓ(23ϵ)Γ(1ϵ)χb(1)χb(3)+2Γ(24ϵ)πΓ(524ϵ)χb(4)+(8ϵ2)Γ(12ϵ)2πΓ(322ϵ)2(χb(2))2

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 cn(p2) coefficient for the potential in [1910.03008, w/ Porto] still holds for log/poles.
  • We explicitly checked the c4(p2) 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