Bridging Positivity and S-matrix Bootstrap Bounds

TL;DR

This work extends the S-matrix Bootstrap to 3+1 dimensions with a concrete EFT interpretation by introducing an effective cutoff, enabling nonperturbative bounds on EFT coefficients c0,c2,c3 and, in O(n) settings, on dimension-six operators. It develops advanced primal and dual formalisms, including a wavelet-inspired amplitude ansatz, subtracted positivity constraints, and linearized unitarity, to map the space of consistent QFTs and isolate EFT-like regions. Key results include a tight bound on c2 in the massless limit, numerical bounds with a mass gap, and dual EFT bounds that connect IR constraints to UV completions; along the boundary, observables such as Spin-zero dominance and UV/IR dominance emerge, clarifying how EFT data encode nonperturbative S-matrix information. The methodology also demonstrates that dimension-six operators in O(n) theories can be bounded nonperturbatively, offering a principled route to connect EFT phenomenology with fundamental S-matrix constraints and informing the interpretation of UV completions.

Abstract

The main objective of this work is to isolate Effective Field Theory scattering amplitudes in the space of non-perturbative two-to-two amplitudes, using the S-matrix Bootstrap. We do so by introducing the notion of Effective Field Theory cutoff in the S-matrix Bootstrap approach. We introduce a number of novel numerical techniques and improvements both for the primal and the linearized dual approach. We perform a detailed comparison of the full unitarity bounds with those obtained using positivity and linearized unitarity. Moreover, we discuss the notion of Spin-Zero and UV dominance along the boundary of the allowed amplitude space by introducing suitable observables. Finally, we show that this construction also leads to novel bounds on operators of dimension less than or equal to six.

Figures (23)

• Figure 1: Analytic properties of the amplitude $M(s,t)$ in the complex $s$-plane for a fixed value of $-4m^2<t^*<4m^2$. We denote in blue the crossing path continuing $M(s,t^*)$ into $M(u,t^*)$. The black dashed vertical line passes through the $s-u$ crossing symmetric point $(4m^2-t^*)/2$. Due to real analyticity, the amplitude is real in between the two cuts and along the dashed line. The right-hand cut is subject directly to the unitarity constraints.
• Figure 2: Allowed values of $(c_0,c_2)$. The different green lines correspond to different $N=6,\dots,12$, while $N=14$ is depicted in blue. The data are obtained at fixed $L=16$: the positivity constraints we impose \ref{['sub_pos']} are so efficient in constraining the large spin behavior that we can consider this value asymptotic.
• Figure 3: On the left plot we show the residue of the pole at threshold $\alpha_\text{th}$ along the boundary of the allowed $(c_0,c_2)$ values in Fig. \ref{['butterfly_plot']}. On the right we show the spin-zero dominance, also along the boundary of the allowed $(c_0,c_2)$ values.
• Figure 4: On the left ratio of derivatives $c_3/c_2$ as a function of the radial angle $\theta$ used to parametrize the boundary of the allowed region in Fig. \ref{['butterfly_plot']}. On the right the same plot in the $(c_2,c_3)$ space, for fixed $N=14$.
• Figure 5: Allowed values of $(c_0,c_2)$ in the un-subtracted physics scenario (left). On the right plot we show the dramatic effect of this constraint: the plot on the left is shown there by a red curve embedded in the plot of Fig. \ref{['butterfly_plot']}.