Mingling of the infrared and ultraviolet and the “cosmological constant” for interacting QFT in 2d

Abstract

We propose a proper definition of the vacuum expectation value of the stress energy tensor 〈0| Tμν| 0〉 for integrable quantum field theories in two spacetime dimensions, which is the analog of the cosmological constant in 4d. For a wide variety of models, massive or massless, we show {rho}_{textrm{vac}}=-{m}^2/2mathfrak{g} exactly, where mathfrak{g} is a generalized coupling which we compute and m is a basic mass scale. The kinds of models we consider are the massive sinh-Gordon and sine-Gordon theories and perturbations of the Yang-Lee and 3-state Potts models, pure Toverline{T} perturbations of infra-red QFT’s, and UV completions of the latter which are massless flows between UV and IR fixed points. In the massive case m is the physical mass of the lightest particle and mathfrak{g} is related to parameters in the 2-body S-matrix. In some examples ρvac = 0 due to a fractional supersymmetry. For massless cases, m can be a scale of spontaneous symmetry breaking. The “cosmological constant problem” generically arises in the free field limit mathfrak{g} → 0, thus interactions can potentially resolve the problem at least for most cases considered in this paper. We speculate on extensions of these results to 4 spacetime dimensions and propose {rho}_{textrm{vac}}={m}^4/2mathfrak{g} , however without integrability we cannot yet propose a precise manner in which to calculate mathfrak{g} . Nevertheless, based on cosmological data on ρvac, if mathfrak{g} ~ 1 then it is worth pointing out that the lightest mass particle is on the order of experimental values of proposed neutrino masses.