KerrEikonal2pm

This is the public data for the paper "Systematic integral evaluation in spin-resummed binary dynamics" arxiv:2406.17658. The Mathematica notebooks provided in this repository are dependent on the package KiHA

Closed form of eikonal

The closed form of the eikonal is stored in "EikonalClosedformList". Summing it up will generate the complete eikonal phase at 2PM in the aligned-spin configuration with $a\cdot b\neq 0$. The output contains structures of the form T[f,rep], where f denotes the function in terms of $a$ and $b$, and rep denotes the replacement rules that define the values of $a$ and $b$ in terms of the spin vector a[1] and the impact parameter b[0]. (For more detail, see the discussions on "sectors" around Eq.(14) in the manuscript.) The placeholder function head T indicates that there are remaining $\sigmai$-integrations to be performed, in order to obtain each term's final contribution to the eikonal phase. Namely, such terms are to be evaluated as $$T[f,rep]\equiv\int0^1 d\sigma1d\sigma2d\sigma3d\sigma4 (f|_{rep}) $$

Spin expansion of the eikonal

To obtain the analytical expression for the aligned-spin eikonal to a given order in spin, the following steps are to be taken: 1. Compute the series expansion of the incomplete elliptic integral integrate[f,{K[1],0,y4}] using the function intSeries[_,{y4,order}]. 2. Substitute the definitions of y2 and y4 in each sector (namely, each T[f, rep]) using the functions Y2[rep] and Y4[rep]. 3. Substitute the values of a and b in each sector as given in rep. 4. Compute the series expansion in powers of a[1]. 5. Integrate over all the $\sigma$ variables in each sector from 0 to 1.

Numerical evaluation of the eikonal exactly in spin

To obtain the numerical value of the aligned-spin eikonal at finite values of spin exactly, the following steps are to be taken: 1. Replace the two incomplete elliptic integrals integrate[f1,{K[1],0,y4}] and integrate[f2,{K[1],0,y4}] with the built-in definitions of the elliptic integrals EllipticF[_,_] and EllipticE[_,_] in Mathematica using the replacement repEKback. 2. Substitute the definitions of y2 and y4 in each sector (namely, each T[f, rep]) using the functions Y2[rep] and Y4[rep]. 3. Substitute the values of a and b in each sector as given in rep. 4. Replace all the dynamic parameters, such as dot[a[1],b[0]], with numerical values 5. Integrate out all the $\sigma_i$ variables from 0 to 1 numerically using NIntegrate

We add an example to illustrate this procedure and show that the spin expansion(orange) converges and agrees with the resummed result(blue) for $|a|/|b|<0.5$. For $|a|/|b|>0.5$, the expansion in spin begins to deviate from the exact spin dependence.

We also note that, in this example, as $|a|/|b|\rightarrow 1$, the branch cut of the square root needs to be chosen explicitly in order to obtain the correct numerical result.

Spin expansion up to $\mathcal{O} (a^8)$

The spin expanded result is stored in the variable EikonalExpandList2

Numerical values of the eikonal for a test Kerr black hole scattering from a background Schwarzschild black hole

The full data of the eikonal is stored in the variable intScalarSpin

In this limit, the Y-diagram contribution is greatly simplified. There are only two integrations, over $\sigma1$ and $\sigma2$, to be carried out. The incomplete elliptic integrals are absent. The eikonal at finite spin is obtained in the same way as above, while the procedure is simplified.