Flux Tube S-matrix Bootstrap

TL;DR

Massless 2D branon S-matrices for flux tubes are constrained by UV completion, linking low-energy phase shifts to Wilson coefficients in the worldsheet EFT. The authors implement a bootstrap using unitarity, crossing and analyticity to bound the nonuniversal K^4 coefficients $\alpha_3$ and $\beta_3$ and translate these bounds into finite-volume energy shifts. They also uncover a resonant structure on the boundary of the allowed S-matrix space, reproducing a worldsheet axion and accompanying states consistent with lattice data. The results offer optimal bounds on irrelevant operators in theories of Goldstones and point toward extensions to higher dimensions and gauge/gravity contexts.

Abstract

We bootstrap the S-matrix of massless particles in unitary, relativistic two dimensional quantum field theories. We find that the low energy expansion of such S-matrices is strongly constrained by the existence of a UV completion. In the context of flux tube physics, this allows us to constrain several terms in the S-matrix low energy expansion or -- equivalently -- on Wilson coefficients of several irrelevant operators showing up in the flux tube effective action. These bounds have direct implications for other physical quantities; for instance, they allow us to further bound the ground state energy as well as the level splitting of degenerate energy levels of large flux tubes. We find that the S-matrices living at the boundary of the allowed space exhibit an intricate pattern of resonances with one sharper resonance whose quantum numbers, mass and width are precisely those of the world-sheet axion proposed in [1,2]. The general method proposed here should be extendable to massless S-matrices in higher dimensions and should lead to new quantitative bounds on irrelevant operators in theories of Goldstones and also in gauge and gravity theories.

Figures (13)

• Figure 1: Allowed region in the $\{\tilde{\gamma}_3,\tilde{\gamma}_5,\tilde{\gamma}_7\}$ space for a generic $D{=}3$ flux tube S-matrix, with $\tilde{\gamma}_n{=}\gamma_n+(-1)^{(n+1)/2}\frac{1}{n 2^{3n-1}}$. The S-matrix at the cusp (black point) is associated to the goldstone (goldstino) S-matrix describing the flow from tricritical Ising to free fermions: it saturates the Schwarz-Pick inequality. The edge in red corresponds to double CDD solutions, saturating the 2-point Schwarz-Pick bound and the full orange surface is determined by the 3-point Schwarz-Pick inequality and it is saturated by a triple CDD family.
• Figure 2: Left: domain of analyticity of a generic massless two-dimensional S-matrix. The cut, in black, is all over the real axis; the threshold at $s=0$ it is in general a branch point singularity. Right: we map the upper half plane to the unit disc through $s\to \chi=(4+is)/(4-is)$. The real axis is mapped to the boundary of the unit circle, the threshold to $\chi=1$ and $s=\infty$ to $\chi=-1$.
• Figure 3: Allowed region in the $\{\beta_3,\alpha_3\}$ parameter space of flux tube S-matrice in $D=4$ as obtained by numerics. The horizontal red line represents the absolute minimum of $\alpha_3$ as predicted analytically by the Schwarz-Pick theorem applied to the symmetric channel. The additional red lines can be obtained applying Schwarz-Pick to the additional crossing symmetric combinations $S_\pm(s)$.
• Figure 4: At the S-matrix space boundary we encounter S-matrices with zeros, that is resonances. In the antisymmetric channel, to the left of the integrable point there is one single resonance while to the right of the integrable point there are two resonances, a broad one and a sharp one. Curiously, as we move along the boundary we encounter S-matrices whose resonance mass and with are in precise agreement with those predicted in Dubovsky:2014fmaDubovsky:2015zey as extracted from $SU(3)$ (right point) and $SU(5)$ (left point) lattice data.
• Figure 5: Allowed space for the first three Taylor coefficients $\{a_0,a_1,a_2\}$ of analytic functions from disk to disk as derived using the Schwarz-Pick lemmas.