Gravitational Causality and the Self-Stress of Photons

Abstract

We study causality in gravitational systems beyond the classical limit. Using on-shell methods, we consider the one-loop corrections from charged particles to the photon energy-momentum tensor - the self-stress - that controls the quantum interaction between two on-shell photons and one off-shell graviton. The self-stress determines in turn the phase shift and time delay in the scattering of photons against a spectator particle of any spin in the eikonal regime. We show that the sign of the $β$-function associated to the running gauge coupling is related to the sign of time delay at small impact parameter. Our results show that, at first post-Minkowskian order, asymptotic causality, where the time delay experienced by any particle must be positive, is respected quantum mechanically. Contrasted with asymptotic causality, we explore a local notion of causality, where the time delay is longer than the one of gravitons, which is seemingly violated by quantum effects.

Figures (5)

• Figure 1: Type of diagram contributing to the eikonal scattering and the resulting time delay via the form factors $F_i$. Curly lines are graviton legs, wiggle lines represent photons, dashed lines are the spectators, and $F_i$ are the form factors defined in Eq. \ref{['formfactorsAll']} associated to the photon energy-momentum tensor.
• Figure 2: Diagrams contributing to the 1-loop discontinuity of the 3-point function with $k^2=k^{\prime\,2}=0$ and $q^2>4m_X^2$ (1-3 crossed triangle diagrams omitted for simplicity). Curly lines are graviton legs, wiggle lines represent photons, charged particles of spin 0, 1/2 and 1 in the loop are represented by $X=\phi,\psi,W$ respectively, and dotted lines put legs that they cut on-shell.
• Figure 3: Integral contour $\Gamma$ in the upper complex $q_1$-plane for $\hat{F}_i$. There are two contributions: one from the graviton pole, and the second from the discontinuity above threshold $t>4 m^2$.
• Figure 4: Quantum corrections to the phase shift as function of $b m$. Dotted lines for large $b m$ show the EFT result which, if allowed to continue to small $bm$, would eventually give a negative total phase shift and thus time delay. See discussion in Sec. \ref{['asymptoticsec']}. Left: We plot the scalar case (as discussed, the spinorial case has similar features, so it is not shown here) that have two contributions coming from the form factors $F_3(t)$, relevant at large impact parameters Eq. \ref{['deltalargeblimit2']}, and $F_1(t)$, which dominates as small $bm$ Eq. \ref{['deltasmallblimit']}. The full numerical solutions Eq. \ref{['eigenvaluesPhaseShift']} is shown as solid lines, and their limiting behaviors as dashed lines. We have taken $\bar{y}=0.27$ to make the approximation close to the exact answer on the scales shown in the plot. Right: We plot the vector-loop case, i.e. QED embedded in a non-abelian gauge theory. The form factors are exponentially suppressed by Sudakov resummation in the region $b m \ll \mathrm{exp}(-\sqrt{\pi/2\alpha})$, as discussed under Eq. \ref{['sudakF1larget']}, but this effect is not displayed here. The vertical blue line represents, for $\alpha=1/100$, the value of $bm$ below which Sudakov resummation can no longer be neglected. For larger values of the impact parameter, $\mathrm{exp}(-\sqrt{\pi/2\alpha}) \ll b m \ll 1$, the fixed-order 1-loop approximation is instead accurate without resummation. In this region, $F_1(t)$ gives the leading contribution to the phase shift Eq. \ref{['deltasmallblimitWNoSudakov']} and it is plotted as a solid red line which interpolates the blue dots representing the exact numerical solution. We have taken $\bar{y}\simeq 1.12$ and $\gamma \simeq 0.12$ in Eq. \ref{['approximationF1']}.
• Figure 5: Quantum corrections to the phase shift in the vector case with IR Sudakov double-logs resummation as a function of $bm$, choosing for simplicity $b_{\mathrm{IR}} = 1/m$. The dots reproduce the full numerical solution for $\alpha=1/10$ (blue), $\alpha=1/100$ (red), and $\alpha=1/200$ (black). The solid lines for $\mathrm{Exp}(-\sqrt{\pi/2\alpha})< bm < 1$ are the analytic approximation based on Eq. \ref{['approximationF1']}. If extrapolated to the region $bm < \mathrm{Exp}(-\sqrt{\pi/2\alpha} )$, the phase shift would turn negative as displayed by the thin dotted lines. In such a regime of small $bm$ the Eq. \ref{['approximationF1']} is however no longer valid. After resumming the IR Sudakov double-logs we find indeed a positive constant phase shift in the region $bm < \mathrm{Exp}(-\sqrt{\pi/2\alpha} )$, and no causality violation.