Positive moments for scattering amplitudes

We find the complete set of conditions satisfied by the forward $2\to 2$ scattering amplitude in unitary 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 a 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 energy and couplings, where the latter has to satisfy two-sided bounds. 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.

Figure 1

The semicircle contour of Eq. (2) in the complex upper ${\widehat{s}}^{\prime }$-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. (20), (21). 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. (23), the red trajectories are examples using Eq. (29) at strong coupling [with the explicit values Eq. (28) and large $g_2$ from Eq. (31)]. Values of $s^4{a}_2/a_0$ larger than the green solid/dashed/dotted lines are excluded by the conditions Eq. (10) 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 $1/180$ with respect to the volume of the unit cube.

Figure 3

In color 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 colors denote points where the distance to the cutoff (i.e., the energy $s_{\max }$ where bounds are saturated) is larger; gray contour lines in the central plot have $s_{\max }/s=2,3,4,\dots$ The black dashed curve denotes the tree-level expectation Eq. (24). 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 $c_{2,1}\ne 0$, 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. [15]). Blue region: allowed by the tree-level $t\ne 0$ bounds of Sec. 3 (for comparison the lower dashed curve reports the results of Ref. [8]). 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. (54); 2D projections in gray. At tree level we have $a_0=c_2$, $a_1=c_4$, ${\partial }_t{a}_0=c_{2,1}$ and ${\partial }_t{a}_1=c_{4,1}$.