Bootstrapping Massive Quantum Field Theories

Abstract

We propose a new non-perturbative method for studying UV complete unitary quantum field theories (QFTs) with a mass gap in general number of spacetime dimensions. The method relies on unitarity formulated as positive semi-definiteness of the matrix of inner products between asymptotic states (in and out) and states created by the action of local operators on the vacuum. The corresponding matrix elements involve scattering amplitudes, form factors and spectral densities of local operators. We test this method in two-dimensional QFTs by setting up a linear optimization problem that gives a lower bound on the central charge of the UV CFT associated to a QFT with a given mass spectrum of stable particles (and couplings between them). Some of our numerical bounds are saturated by known form factors in integrable theories like the sine-Gordon, E8 and O(N) models.

Figures (12)

• Figure 1: Contribution of the one particle states of the second breather into the UV central charge as a function of $m_2^2$. On the horizontal axis the mass $m_2$ is given in the units of $m_1$. The range of parameters which allow for the existence of the second breather is provided in \ref{['eq:parameters_2breathers']}. For $m_2^2=3$ we have $c_1\approx 0.72126$.
• Figure 2: Contribution of the two particle states to the total central charge as a function of $N$. Dots represent the numerical values and the solid line represents the best linear fit applied for points with $N\geq 20$ only. The values of $c_2$ for small values of $N$ differ notably from the linear asymptotics. For examples for $N=3, 4, 5$ we have $c_2\approx 1.6,\; 2.39,\; 3.26$.
• Figure 3: Lower bound on the UV central charge as a function of the cubic coupling $g$ between particles of mass $m_1=1$, $m_1=1$ and $m_2=\sqrt{3}$. The allowed region is depicted in blue. The bound extends up to $g=4.55901$ which is a critical value for which the optimization problem is feasible. The bound was obtained with $N_{max}=50$. The red horizontal line at $c=1/2$ is added for convenience.
• Figure 4: The real part, the imaginary part and the absolute value of the partial amplitude for the scattering of the lightest asymptotic state with $m_1=1$ given the mass of the second asymptotic state $m_2=\sqrt{3}$ and the value of the trilinear coupling \ref{['eq:critical_g']}. The plot is constructed with $N_{max}=50$.
• Figure 5: The real and imaginary parts of the two particle form factor for the masses $m_1=1$ and $m_2=\sqrt{3}$ and the value of the trilinear coupling \ref{['eq:critical_g']}. The plot is constructed with $N_{max}=50$.