Positive Moments for Scattering Amplitudes

TL;DR

This work develops a complete positivity framework for forward 2→2 scattering amplitudes in EFTs by introducing arc variables $a_n(\hat{s})$ that are dispersive moments of the forward amplitude. Through a Hausdorff-moment analysis, it derives necessary and sufficient positivity constraints, and provides optimal finite-N bounds expressed via Hankel-positivity conditions that bound Wilson coefficients in terms of a finite number of arcs. It then examines how RG running and finite-$t$ corrections modify these bounds, demonstrating that supersoft amplitudes cannot be UV-completed under the studied assumptions and identifying regimes where tree-level arcs suffice versus those where quantum effects dominate. The approach yields a practical, scale-aware toolkit to constrain EFT parameters and infer UV resonance structure from low-energy data, with potential extensions toward non-forward kinematics and S-matrix bootstrap methods.

Abstract

We find the complete set of conditions satisfied by the forward $2\to2$ scattering amplitude in unitarity and causal theories. These are based on an infinite set of energy dependent quantities -- the arcs -- which are dispersively expressed as moments of a positive measure defined at (arbitrarily) higher energies. We identify optimal finite subsets of constraints, suitable to bound Effective Field Theories (EFTs), at any finite order in the energy expansion. At tree-level arcs are in one-to-one correspondence with Wilson coefficients. We establish under which conditions this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss the range of validity in both couplings and energy. We also extend our results to the case of small but finite~$t$. A consequence of our study is that EFTs in which the scattering amplitude in some regime grows in energy faster than $E^6$ cannot be UV-completed.

Figures (5)

• Figure 1: The semi-circle contour of Eq. (\ref{['archdef']}) in the complex upper ${\hat{s}^\prime}$-plane. Wiggle lines denote the branch-cuts on the real axis.
• Figure 2: Allowed regions for the arcs $a_0$, $a_1$, $a_2$ (LEFT) and including $a_3$ (RIGHT), according to Eqs. (\ref{['a2space']},\ref{['a3space']}). LEFT: For fixed Wilson coefficients, as energy is increased, the theory spans a trajectory in the space of arcs: the blue trajectories (arrows in the direction of increasing $s$) correspond to examples in the weak coupling limit Eq. (\ref{['tree1']}), the red trajectories are examples using Eq. (\ref{['arcsgoldstone']}) at strong coupling (with the explicit values Eq. (\ref{['explicitgoldstones']}) and large $g_2$ from Eq. (\ref{['couplings']})). Values of $s^4a_2/a_0$ larger than the green solid/dashed/dotted lines are excluded by the conditions Eq. (\ref{['cc']}) from Bernstein polynomials, up to $k+n=2,3,4$ respectively. RIGHT: The projections into two dimensional planes correspond to optimal bounds when only two coefficients are taken into account (the bottom projection corresponds to the left panel). The volume of the allowed region is $1/180$ w.r.t. the volume of the unit cube.
• Figure 3: In colour the allowed area for (combinations of) Wilson coefficients $c_2$, $c_4(s)$ and $c_6(s)$, evaluated at a scale $s$. In all panels $\beta_4/c_4(s)=0.1$. Left and center panels: warmer colours denote points where the distance to the cutoff (i.e. the energy $s_{max}$ where bounds are saturated) is larger; grey contour lines in the central plot have $s_{max}/s=2,3,4,\ldots$ The black dashed curve denotes the tree-level expectation Eq. (\ref{['3arcstreelevel']}). The region above the black lines have $\beta_4/c_6(s_{max})s_{max}^2<0.1,0.2,0.4$ respectively. The inset in the center panel shows a wider region of parameter space. Right panel: the same as in center panel but with logarithmic scale.
• Figure 4: The Galileon case $c_{2,1}\neq 0$, for fixed ${-\beta_4/c_4(s)=0.1}$. Orange Region: allowed from bounds on RGE of the forward $(t=0)$ amplitude (for comparison the upper dashed curve shows the upper bound from Ref. Bellazzini:2017fep). Blue region: allowed by the tree-level $t\neq0$ bounds of section \ref{['sec:bf']} (for comparison the lower dashed curve reports the results of Ref. deRham:2017avq). The solid black curve shows the intersection. Notice that upper and lower parts of the plot have different scales.
• Figure 5: Allowed region in the space of arcs and their first $t$-derivative for ${\partial_t a_0<0}$, according to Eq. (\ref{['finitetsoftkiller']}); 2D projections in grey. At tree level we have $a_0=c_2$, $a_1=c_4$, $\partial_ta_0=c_{2,1}$ and $\partial_ta_1=c_{4,1}$.