Selection rules for the S-Matrix bootstrap

TL;DR

The paper investigates the space of allowed pion-pion S-matrices using a numerically intensified S-matrix bootstrap, imposing crossing symmetry, Adler zeros, unitarity, and a modeled $\rho$ resonance to reveal lake and river regions. It identifies two Adler-zero regions, A and C, where linear Regge trajectories emerge and intersect the Lovelace-Shapiro line, with A notably aligning with a kink near the two-loop $\chi$PT$ values. By examining the averaged total cross section $\bar{\sigma}$ and averaged entanglement power $\bar{\mathcal{E}}$, the authors propose selection rules that correlate reduced cross sections and entanglement with linear Regge behavior, isolating A and C as physically interesting regions and linking them to experimental S- and P-wave scattering lengths. The work also demonstrates a method to trace a minimum-$s_2$ curve along the river, finding a curve that passes near A and C and exhibits Regge behavior, while hypothesis testing favors region A as closer to $\chi PT$ predictions; overall, the study suggests a nonperturbative, information-theoretic mechanism for selecting physically relevant S-matrices with potential ties to string theory and holographic ideas.

Abstract

We examine the space of allowed S-matrices on the Adler zeros' plane using the recently resurrected (numerical) S-matrix bootstrap program for pion scattering. Two physical quantities, an averaged total scattering cross-section, and an averaged entanglement power for the boundary S-matrices, are studied. Emerging linearity in the leading Regge trajectory is correlated with a reduction in both these quantities. We identify two potentially viable regions where the S-matrices give decent agreement with low energy S- and P-wave scattering lengths and have leading Regge trajectory compatible with experiments. We also study the line of minimum averaged total cross section in the Adler zeros' plane. The Lovelace-Shapiro model, which was a precursor to modern string theory, is given by a straight line in the Adler zeros' plane and, quite remarkably, we find that this line intersects the space of allowed S-matrices near both these regions.

Figures (14)

• Figure 1: Pion river at $N_{max}=16$. The behaviour changes rapidly at points A,B,C and D. The red cross marks the two-loop $\chi PT$. The brown straight line is the Lovelace-Shapiro model allowing for general Adler zeros. The green regions marks exhibit linearity. The inset shows a zoomed version with the tree-level, one-loop and two-loop $\chi PT$ values indicated in black, orange and red respectively.
• Figure 2: (a) Variation of best fit line with $s_0$ on the upper boundary. For $s_0=0.35$ the best fit line including $\ell=0$ to $\ell=6$ is given by $J=0.38 + 0.51 s_R$ while the experimental one is $J=0.27 +0.54 s_R$. (b) Variation of the slope $\alpha'$ with $s_0$ on the upper boundary in the neighbourhood of ${\bf A}$. Except near $s_0\approx 0.35$ the even/odd spins separate.
• Figure 3: (a) Variation of $\bar{\mathcal{E}}$ and $\bar{\sigma}$ with $s_0$ for $s_{cut}=375$ for upper boundary. (b) Variation of $\bar{\mathcal{E}}$ and $\bar{\sigma}$ with $s_0$ for the lower boundary for $s_{cut}=375$. Blue line is for $\bar{\sigma}^{\pi^0\pi^0}$, black for $\bar{\sigma}^{\pi^+\pi^-}$ and red for $\bar{\mathcal{E}}$. The green bar indicates linear Regge trajectory with $R^2>0.93$.
• Figure 4: (a) The black curve within the river river gives the $s_2$ with minimum average cross section for each $s_0$. Note that the curve passes very close to regions A and C described earlier (b) The value of the corresponding minimum as a function of $s_0$ for $N_{max}=14$ and $L_{max}=19$. The global minimum average cross section appears to be at $s_0=4$
• Figure 5: (a) Variation of best-fit $R^2$ for the S-matrices on the minimum line. Only even spins are considered. An average $R^2>0.95$ shows that all such S-matrices show good linearity (b) Variation of best-fit slope($\alpha'$) of the S-matrices as a function of $s_0$.