Spinning S-matrix Bootstrap in 4d

TL;DR

The paper extends the numerical S-matrix bootstrap to four-dimensional theories with spinning particles, formalizing a complete framework for unitarity and crossing of 2→2 amplitudes with spin. It develops COM- and covariant tensor approaches, introduces improved kinematic-analytic amplitudes to remove nonphysical singularities, and applies the method to identical Majorana fermions to extract nonperturbative bounds. Using a primal semidefinite program, the authors bound the quartic coupling and Yukawa-type couplings, obtaining numerically robust limits that are consistent with naive dimensional analysis and perturbative checks. The work sets the stage for broader spinning-S-matrix bootstrap explorations (e.g., photons, gluons, pions) and demonstrates a practical path to constraining EFTs nonperturbatively via S-matrix consistency alone.

Abstract

We review unitarity and crossing constraints on scattering amplitudes for particles with spin in four dimensional quantum field theories. As an application we study two to two scattering of neutral spin 1/2 fermions in detail. Assuming Mandelstam analyticity of its scattering amplitude, we use the numerical S-matrix bootstrap method to estimate various non-perturbative bounds on quartic and cubic (Yukawa) couplings.

Figures (12)

• Figure 1: Upper bound on the quartic coupling $\lambda$ as a function of $N_{max}^{-1}$. The numerical results are indicated by the red dots. They correspond to $N_{max}=10, 16,18,20,22$ and $24$. The blue line represents the linear fit of the three points $N_{max}=20,\,22$ and $24$. It is described by $\lambda/(32\pi) = 1.74 - 6.02\, N_{max}^{-1}$ equation. The spin cut-off parameter used here is $L_{max}=N_{max}+20$.
• Figure 2: Lower bound on the quartic coupling $\lambda$ as a function of $N_{max}^{-1}$. The numerical results are indicated by the red dots. They correspond to $N_{max}=10, 16,18,20,22$ and $24$. The blue line represents the linear fit of the three points $N_{max}=20,\,22$ and $24$. It is described by $\lambda/(32\pi) = -3.25 + 13.76\, N_{max}^{-1}$ equation. The spin cut-off parameter used here is $L_{max}=N_{max}+20$.
• Figure 3: Upper bound on the quartic coupling $\lambda$ as a function of $L_{max}^{-1}$. The dots represent the numerical results. The solid lines represent the linear extrapolation in $L_{max}^{-1}$ based on the last four points for each $N_{max}$. Different colours correspond to different values of $N_{max}$ indicated in the right-hand side of the plot.
• Figure 4: Lower bound on the quartic coupling $\lambda$ as a function of $L_{max}^{-1}$. The dots represent the numerical results. The solid lines represent the linear extrapolation in $L_{max}^{-1}$ based on the last four points for each $N_{max}$. Different colours correspond to different values of $N_{max}$ indicated in the right-hand side of the plot.
• Figure 5: Upper bound on the cubic Yukawa coupling $g$ as a function of the scalar particle mass $M$. The bound is constructed for $N_{max}=12,\,16,\,20,\,24$ and $L_{max}=N_{max}+20$. Using $N_{max}=20$ and $N_{max}=24$ we also perform a linear extrapolation of the bound to $N_{max}=\infty$. The latter is indicated by the dashed line.