[Paper Review] Exponentiation of the leading eikonal with spin

Kays Haddad|arXiv (Cornell University)|Sep 9, 2021

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper demonstrates that the leading eikonal phase for gravitational scattering of spin-1/2 particles exponentiates up to $Ó(G^2)$ in the $D\to 4$ limit, even with arbitrary spin orientations. Using unitarity at leading order in $\hbar$, the authors show that the square of the one-loop eikonal matches super-classical divergences, confirming exponentiation in impact-parameter space for spinning systems.

ABSTRACT

We initiate a study into the eikonal exponentiation of the amplitude in impact-parameter space when spinning particles are involved in the scattering. Considering the gravitational scattering of two spin-1/2 particles, we demonstrate that the leading eikonal exhibits exponentiation up to $\mathcal{O}(G^{2})$ in the limit where the spacetime dimension $D ightarrow4$. We find this to hold for general spin orientations. The exponentiation of the leading eikonal including spin is understood through the unitarity properties at leading order in $\hbar$ of momentum-space amplitudes, allowing the extension of our results to arbitrary-spin scattering.

Motivation & Objective

• To investigate whether the leading eikonal phase exponentiates in impact-parameter space when spinning particles are involved in gravitational scattering.
• To resolve the gap in understanding exponentiation of the eikonal phase in the presence of massive spinning matter, particularly at higher post-Minkowskian orders.
• To establish a connection between super-classical divergences in one-loop amplitudes and the square of the leading eikonal phase in the $D\to 4$ limit.
• To generalize the exponentiation mechanism via unitarity relations to arbitrary-spin scattering, extending results from spin-1/2 systems.

Proposed method

• The authors compute the leading eikonal phase for spin-1/2 particle scattering up to $\mathcal{O}(G^2)$ in the $D\to 4$ spacetime limit.
• They relate the square of the one-loop eikonal to the super-classical divergences of the one-loop amplitude using momentum-space unitarity at leading order in $\hbar$.
• The analysis employs the Fourier transform of the $2\to 2$ amplitude in impact-parameter space to define the eikonal phase via $\frac{1}{\hbar}\delta_n(\mathbf{b}) = \widetilde{\mathcal{M}}_n^{\text{cl.}}(\mathbf{b})$.
• They use the convolution structure of one-loop amplitudes in momentum space, showing that the product of two $2\to 2$ amplitudes maps to a product in impact-parameter space.
• The derivation relies on energy and momentum conservation via delta functions, with $k_{1,2}^*$ roots used to simplify the energy delta function integral.
• The method generalizes from spin-1/2 to arbitrary spin by exploiting unitarity properties of amplitudes at $\mathcal{O}(\hbar^0)$, ensuring consistency across spin types.

Experimental results

Research questions

• RQ1 Does the leading eikonal phase exponentiate in impact-parameter space when spin-1/2 particles scatter gravitationally in the $D\to 4$ limit?
• RQ2 How are super-classical divergences in the one-loop amplitude related to the square of the leading eikonal phase?
• RQ3 Can the exponentiation mechanism be generalized beyond spin-1/2 to arbitrary-spin particles using unitarity at leading order in $\hbar$?
• RQ4 What role does the $D\to 4$ limit play in enabling the exponentiation of the eikonal phase with spin?
• RQ5 Is the exponentiation of the eikonal phase consistent across different spin orientations in the scattering process?

Key findings

• The leading eikonal phase for spin-1/2 particle scattering exponentiates up to $\mathcal{O}(G^2)$ in the $D\to 4$ limit, confirming the exponentiation structure in impact-parameter space.
• The square of the one-loop eikonal phase matches the super-classical divergences of the one-loop amplitude, providing a direct link between eikonal exponentiation and quantum corrections.
• Exponentiation holds for arbitrary spin orientations, indicating robustness of the mechanism beyond aligned-spin configurations.
• The unitarity properties of momentum-space amplitudes at leading order in $\hbar$ underlie the exponentiation, enabling generalization to arbitrary-spin systems.
• The convolution of two $2\to 2$ amplitudes in momentum space maps to a product in impact-parameter space, validating the eikonal exponentiation framework.
• The result supports the conjecture that the exponentiated eikonal phase can generate classical observables such as scattering angle and spin kick, even at $\mathcal{O}(G^3)$ and beyond.