Vacuum energy density from the form factor bootstrap

TL;DR

This work develops a prescription to compute the vacuum energy density $\rho_{\rm vac}$ from the form-factor bootstrap by relating the zero-particle form factor of the trace $\Theta$ to the two-particle sector. In $D=2$ integrable QFTs, the authors derive a precise high-energy limit of the crossed $2$-particle form factor that yields $\langle 0|\Theta|0\rangle$ and reproduces known results from the thermodynamic Bethe Ansatz, avoiding finite-temperature methods. They propose a natural generalization to $D=4$, predicting the universal scaling $\rho_{\rm vac} \propto m^{4}/\mathfrak{g}$ with the dimensionless coupling $\mathfrak{g}$ fixed by the asymptotic S-matrix, giving $\rho_{\rm vac} = \dfrac{3}{4}\dfrac{m^{4}}{\mathfrak{g}}$. The discussion connects this framework to the cosmological constant problem, suggesting a UV-origin particle dubbed the zeron (potentially a Majorana neutrino) and outlining possible experimental and theoretical implications for physics beyond the Standard Model.

Abstract

The form-factor bootstrap is incomplete until one normalizes the zero-particle form factor. For the stress energy tensor we describe how to obtain the vacuum energy density $ρ_{\rm vac}$, defined as $\langle 0| T_{μν} | 0 \rangle = ρ_{\rm vac} \, g_{μν}$, from the form-factor bootstrap. Even for integrable QFT's in D=2 spacetime dimensions, this prescription is new, although it reproduces previously known results obtained in a different and more difficult thermodynamic Bethe ansatz computation. We propose a version of this prescription in D=4 dimensions. For these even dimensions, the vacuum energy density has the universal form $ρ_{\rm vac} \propto m^D/\mathfrak{g}$ where $\mathfrak{g}$ is a dimensionless interaction coupling constant which can be determined from the high energy behavior of the S-matrix. In the limit $\mathfrak{g} \to 0$, $ρ_{\rm vac} $ diverges due to well understood UV divergences in free quantum field theories. If we assume the the observed Cosmological Constant originates from the vacuum energy density $ρ_{\rm vac}$ computed as proposed here, then this suggests there must exist a particle which does not obtain its mass from spontaneous symmetry breaking in the electro-weak sector, which we designate as the "zeron". A strong candidate for the zeron is a massive Majorana neutrino.