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 $⟨0|T_{\mu \nu }|0⟩$ 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 ${\rho }_{\mathrm{vac}}=-m^2/2\mathfrak{g}$ 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 $T\bar{T}$ 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 mass of the lightest particle and $\mathfrak{g}$ is related to parameters in the 2-body S-matrix. In some examples ${\rho }_{\mathrm{vac}}=0$ due to an unbroken 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 $\mathfrak{g}\to 0$, 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 ${\rho }_{\mathrm{vac}}=-m^4/2\mathfrak{g}$,however without integrability we cannot yet propose a precise manner in which to calculate $\mathfrak{g}$. Nevertheless, based on cosmological data on ${\rho }_{\mathrm{vac}}$, if $\mathfrak{g}\approx 1$ then the lightest mass particle is on the order of experimental values of proposed neutrino masses.

Comments: 10 pages; version 2: The only change is our convention for the sign of the vacuum energy ${\rho }_{\mathrm{vac}}$, which is now more in line with standard 4d cosmology literature version3: We included the example of the sine-Gordon model which is very illustrative. Previous results didn't change. Version 4: published version in JHEP, primary change is added references