S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

TL;DR

The paper advances the S-matrix bootstrap in $3+1$ dimensions by formulating a primal conic optimization for $2\rightarrow2$ scattering under analyticity, crossing, and unitarity, and introducing a regularized dual convex problem with free dual partial waves $k_\ell(s)$. It shows that the dual variables correctly bound the primal maximum and relate to physical partial waves via $h_\ell(s) = -i\left(\frac{k_\ell(s)}{|k_\ell(s)|}-1\right)$, enabling upper bounds without needing a specific pole structure a priori. Generalized dispersion relations yield a two-variable dual amplitude in 3+1d that encodes the same physics as the primal amplitude, allowing extraction of partial waves from the dual solution. Numerical experiments in the scalar (pion-like) case demonstrate sharp bracketing of the maximum and consistency with known results, even without assuming a threshold pole, illustrating a practical approach to bracket the space of allowed S-matrices with a small set of dual variables.

Abstract

The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the $2\rightarrow 2$ scattering matrix $S_{2\rightarrow 2}$ such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider $3+1$ dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves $k_\ell(s)$ that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves $f_\ell(s)$, for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.

Figures (6)

• Figure 1: The maximum of the amplitudes along $s_0=t_0$ in the (large) Mandelstam triangle. The green points are the dual (upper bounds), the red are the primal (lower bounds for the maximum). Both were run without assuming the existence of a pole at threshold. The blue line represents the primal problem once a pole at threshold is included. The black diamonds represent the pioneering results of Lopez and Mennessier LM. The inset shows more clearly the results near the symmetric point which is the minimum of the blue curve where $\frac{\pi}{4}F(\frac{4}{3},\frac{4}{3},\frac{4}{3})\simeq2.6613$. The factor $\pi/4$ is to match the normalization in the literature, e.g.Paulos:2017fhb.
• Figure 2: Mandelstam diagram in the real ($s$, $t$, $u$) plane. The red regions are the Mandelstam regions where the amplitude spectral densities $\rho^{st}$ and $\rho^{su}$ in \ref{['a106']} have support. The green regions are the physical regions in the $s$, $t$ and $u$ channels where the dual amplitudes \ref{['Kdualsp']} and \ref{['KdualON']} have double jumps. The triangles in the middle are the Mandelstam triangles where we usually evaluate the amplitude for maximization and the diagonal blue line $s_0=t_0$ is where we maximize the amplitude to produce fig.\ref{['fig3']}.
• Figure 3: The maximum of the amplitude at the symmetric point $s_0=t_0=u_0=\frac{4}{3}$ as the regulator $M_{\hbox{reg}}$ is increased for the primal (red) and dual (green) problems. The horizontal axis gives the regulator in a logarithmic scale and up to a horizontal shift to plot the curves together. As explained in the main text, the primal improves by increasing the number of interpolation points, as indicated with the red upward arrow, and the dual improves by increasing the number of coefficients, as indicated with the green downward arrow. We do not asume the existence of a pole at threshold which would give the horizontal blue line. The values are multiplied by a factor $\pi/4$ to match the normalization in the literature, e.g.Paulos:2017fhb.
• Figure 4: The maximum of the amplitude at the symmetric point $s_0=t_0=u_0=\frac{4}{3}$ as the (inverse of the) number of interpolation points/coefficients $N^{-1}_{\text{coeff}}$ changes for the primal (red) and the dual (green). The vertical axis correspond to infinite number of interpolation points for the primal or coefficients for the dual. We do not assume the existence of a pole at threshold which would give the horizontal line. The values are multiplied by a factor $\pi/4$ to match the normalization in the literature, e.g. see Paulos:2017fhb.
• Figure 5: Real and imaginary parts of the S-wave $h_0(s)$ for the amplitude from maximizing at the symmetry point. The blue curve is the primal result by imposing a threshold pole, and the green circles are the dual result using the expression \ref{['c55']}without assuming the threshold pole. The two agree perfectly.