Effective Field Theory Calculation of LIGO-like Compton Scattering

Abstract

We use effective field theory (EFT) to calculate the scattering amplitude of a LIGO-like graviton-scalar Compton interaction. We gauge the center-of-momentum energy $\sqrt{s}$ between one gravitational-wave (GW) graviton (one quantum of the coherent bulk of an astrophysical GW, with energy $E_g=\hbarω_\mathrm{GW}$) and a resting heavy target (a suspended mass with rest energy $E_M=m_Mc^2$ found in laser interferometer observatories) to be of order $\sim10^{1.5}$ PeV -- at the energy scale within the extremes of astroparticle physical phenomena. This back-of-the-envelope calculation supports the calculation of a convergent cross section in our LIGO-like Compton analysis, which we indeed recover using standard EFT Feynman rules and relevant traceless-transverse gauges for the graviton polarizations. We obtain that the cross section $σ$ is largely dependent on the center-of-momentum energy, and from this we define the corresponding impact parameter via $σ=πb^2$. This impact parameter, after coherence-state population enhancement, scales with GW energetics along with the unique coupling (in natural units) $\sqrt{ω_\mathrm{GW}/m_M}\sim10^{-21}$ -- the same order of magnitude as astrophysical GW strain. One finds furthermore that, after isolating the GW energetics sector from the GW-mirror impact parameter, the revised length scale $\tilde{b}/(GM)\approx1.76π$ quantifies the pre-merger stage of compact binary coalescence, which is compared with $b/(GM)>14$ calculated from the early-inspiral worldline quantum field theory framework. Conventional insight of GW-mirror response is recovered, such that the impact parameter $b$ scales exactly with the mirror recoil $ΔL\sim10^{-18}\,\mathrm{m}$ after having made contact with the GW.