The S-matrix Bootstrap IV: Multiple Amplitudes

TL;DR

This work advances the S-matrix bootstrap program in two dimensions by analyzing a Z2-symmetric system with two stable particles, leveraging both a multi-amplitude S-matrix approach and a multi-correlator AdS/CFT bootstrap to bound the cubic couplings $g_{112}$ and $g_{222}$. The authors develop a detailed kinematic and unitarity framework, implement a dispersion-relation based semidefinite program, and uncover a rich structure in the allowed coupling space, including integrable points such as the Potts model, SUSY-Sine-Gordon, and an elliptic deformation line. They show strong cross-consistency between the S-matrix and AdS bootstrap results, particularly near equal masses, and expose phenomena like extended unitarity and screening that shape the boundary of the coupling space. The work points to exciting future directions, including higher dimensions, more particles, and deeper connections between integrable S-matrices and their AdS/CFT counterparts.

Abstract

We explore the space of consistent three-particle couplings in $\mathbb Z_2$-symmetric two-dimensional QFTs using two first-principles approaches. Our first approach relies solely on unitarity, analyticity and crossing symmetry of the two-to-two scattering amplitudes and extends the techniques of [arXiv:1607.06110] to a multi-amplitude setup. Our second approach is based on placing QFTs in AdS to get upper bounds on couplings with the numerical conformal bootstrap, and is a multi-correlator version of [arXiv:1607.06109]. The space of allowed couplings that we carve out is rich in features, some of which we can link to amplitudes in integrable theories with a $\mathbb Z_2$ symmetry, e.g., the three-state Potts and tricritical Ising field theories. Along a specific line our maximal coupling agrees with that of a new exact S-matrix that corresponds to an elliptic deformation of the supersymmetric Sine-Gordon model which preserves unitarity and solves the Yang-Baxter equation.

Figures (29)

• Figure 1: Diagonal processes are those where the incoming and outgoing particles have the same momenta as illustrated in the first row; they are all crossing invariant. The non-diagonal processes in the second row are those for which the final momenta are not the same as the initial momenta. Swapping space and time interchanges the odd and even off-diagonal processes so these off-diagonal processes play a crucial role in connecting these two sectors of different $\mathbb{Z}_2$ charge.
• Figure 2: Upper bounds on the cubic coupling $g_{112}^2$ as a function of $\mu\equiv m_2/m_1$. Dashed line: Analytic bound based on the scattering of the lightest odd particle, from Paper2. Solid line: Analytic bound arising from the forward (or transmission) scattering of the odd particle against the even particle; it is a much stronger bound. Red dots: The numeric bound obtained from all two-to-two processes as discussed in the main text. The shaded regions represent the allowed regions which nicely shrink as we include more constraints. Any relativistic, unitary, $\mathbb{Z}_2$ invariant theory theory with two stable particles (one odd with mass $m_1$ and one even with mass $m_2$) must lie inside the darkest blue region.
• Figure 3: Upper bounds on the cubic coupling $g_{222}$ as a function of $\mu\equiv m_2/m_1$. Solid line: Analytic bound based on the scattering of the lightest even particle, from Paper2. Red dots: The numeric bound obtained from all two-to-two processes as discussed in the main text. The shaded region represent the allowed region. When the even particle is the lightest, we can solve analytically for the maximal coupling, even considering the full set of amplitudes. When the odd particle is the lightest, the coupling can be bigger, diverging when singularities of the amplitudes corresponding to physical processes collide. This happens at $m_2/m_1 = 2/\sqrt{3}$. After this mass ratio the upper bound disappears.
• Figure 4: Analytic structure of the $S_{22\to 22}$ amplitude (for clarity we do not show the left cut and s-channel pole following from crossing symmetry $S_{22\to 22}(s)=S_{22\to 22}(4m_2^2-s)$). If $m_2$ is not the lightest particle, there is a new feature in the $S_{22\to 22}$ amplitude: a two particle cut starting at $s=4m_1^2$ corresponding to the contribution of two particles $m_1$. This cut appears before the cut for two particles $m_2$ at physical energies $s \ge 4m_2^2$ where regular unitarity is imposed and the amplitude needs to be bounded. As $m_2$ grows beyond $2/\sqrt{3} m_1$ the t-channel pole corresponding to the exchange of particle $m_2$ enters the new cut (by crossing symmetry the $s$-channel pole enters the $t$-channel cut) so we "lose" this pole. Beyond this point we can no longer bound $g_{222}$ since it does not appear in any other diagonal amplitude. This is indeed what we observe in the numerics as illustrated in figure \ref{['g222']}. Note that before the bound on $g_{222}$ disappears it diverges. This divergence, arising from the collision of the $t$-channel pole with an $s$-channel cut is analogous to the divergences in bounds on couplings when $s$-- and $t$-- channel poles collide as already observed in Paper2; the dashed line in figure \ref{['g112']} which was taken from Paper2 diverges at $m_2=\sqrt{2} m_1$ for exactly this reason.
• Figure 5: In two dimensions when we scatter two particles $m_a$ and $m_b$ from the infinite past with $m_a$ to the left of $m_b$ we can end up, in the infinite future with $m_a$ to the right of $m_b$ or vice-versa. If the particles are distinguishable these are two genuinely different processes denoted as the forward or backward process. (They are sometimes also called the transmission and reflection processes.) In higher dimensions, these two scenarios are limiting values of the a single amplitude when the scattering angle tends to $\theta=0$ or $\theta=\pi$, but in two dimensions there is no scattering angle and these processes are described by independent functions. As we exchange time and space, i.e. as we analytically continue these processes by swapping $t$ and $s$ we see that the forward process is mapped to itself while the backward process as seen from its crossed channel describes the $m_a m_a \to m_b m_b$ event. This translates into equations (\ref{['C1']}) and (\ref{['C2']}) in the main text.

Theorems & Definitions (1)

• Lemma 1