Non-Forward UV/IR Relations

TL;DR

This work develops a forward-limit–free program to derive positivity bounds on EFT Wilson coefficients in theories with massless particles and gravity by employing non-forward dispersion relations defined at finite momentum transfer. It introduces smeared dispersion relations and a systematic finite-t improvement algorithm that cancels higher UV contributions, reducing the problem to a semidefinite program over a finite basis (e.g., Legendre polynomials) to bound coefficients like $g_{2,0}$, $g_{3,1}$, and $g_{4,0}$. The analysis covers both strictly tree-level and loop-corrected amplitudes, clarifying how gravity induces non-analyticities that invalidate Taylor expansions near $t=0$ and how the finite-t approach remains robust by treating loop effects perturbatively. The results demonstrate convergence to known forward-limit bounds in the absence of gravity, quantify gravity-induced relaxations, and provide a practical framework for extracting UV-reliability constraints in EFTs with massless particles.

Abstract

We study bounds arising from the analyticity and unitarity of scattering amplitudes in the context of effective field theories with massless particles. We provide an approach that only uses dispersion relations away from the forward limit. This is suitable to derive constraints in the presence of gravity, in a way that is robust with respect to radiative corrections. Our method not only allows us to avoid the Coulomb pole, but also the singularities associated with calculable loop effects, which would otherwise diverge.

Figures (9)

• Figure 1: The $2\rightarrow2$ scattering amplitude is analytic in the upper (and by crossing, lower) half plane in $s'$. The contour in red is the IR arc Eq. (\ref{['eq:archdeft']}) that runs within the EFT validity region, whereas the contour in blue denotes the arc in the UV Eq. (\ref{['eq:archdeftUV']}). In magenta the IR part of the branch cut, due to loops in the EFT, in black its UV part, associated also to particle exchange in the unknown UV theory. We indicate the two subtraction points at $s'=0$ and $s'=-t$.
• Figure 2: Bounds on the ratio $g_{3,1}s/g_{2,0}$ as a function of $t_{\text{max}}$, the maximum value of $t$ in the smearing integral. All curves are derived with fixed $j_\text{max}=7$, $\ell_\text{max}=14$. In blue the improvement approach of Ref. Caron-Huot:2021rmr where positivity is obtained expanding the integrand at order $O(t_\text{max}^{16})$. In purple (orange), our improvement at order $N=3$ ($N=8$). The dashed vertical line delimits the applicability of our improvement formula, $t_\text{max}\leq t_*$. As $N$ is increased, the algorithm at finite $t$ of Section \ref{['sec:finitetimprovement']} rapidly converges to the value of Ref. Caron-Huot:2021rmr.
• Figure 3: Positivity bounds on the ratio of coefficients $g_{4,0}/g_{2,0}$ and $g_{3,1}/g_{2,0}$ in units of $s$, for $d=6$, using $\ell_\text{max}=100$ and $j_\text{max}=10$. In blue the bounds using our partial improvement Eq. (\ref{['eq:UVimpgenC']}) with $N=6$ and $t_{\text{max}}=t_*/2\approx 0.2 s$. The approach of Ref. Caron-Huot:2021rmr is shown for comparison in black for $t_{\text{max}}=s$, in red for $t_{\text{max}}=0.8s$ and orange for $t_{\text{max}}=0.5s$.
• Figure 4: Bounds in the presence of gravity on the ratios $g_{2,0}s/\kappa^2$ and $g_{3,1}s^2/\kappa^2$, in $d=6$, for different values of $t_{\rm max}$. All bounds in this figure are derived using the improvement of Ref. Caron-Huot:2021rmr. The blue (orange) line uses $j_\text{max}=6\,\,(9)$, and is expanded up to order $16\,\,(25)$ in $t_\text{max}$.
• Figure 5: Same as Figure \ref{['fig:gravtmax']}, but using our improvement formula Eq. (\ref{['eq:UVimpgenC']}) up to order $N=12$ (refer to Table \ref{['tab:consistentparams']} and \ref{['tb:values']} for the exact choices of parameters we make). All the bounds are computed with $\ell_\text{max}=400$, and up to $j_\text{max}$=8. Here $t_{\max}$ is expressed in units of the maximum allowed smearing range $t_*\approx 0.39 s$. For reference, in black we show the bounds from Ref. Caron-Huot:2021rmr.