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We propose a proper definition of the vacuum expectation value of the stress
energy tensor $\langle 0 | T_{\mu\nu} |0 \rangle$ 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_{\rm 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 $\rho_{\rm vac} =0$ 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
$\rho_{\rm vac} =- m^4/2 \mathfrak{g}$,however without integrability we cannot
yet propose a precise manner in which to calculate $\mathfrak{g}$.
Nevertheless, based on cosmological data on $\rho_{\rm vac} $, if $\mathfrak{g}
\approx 1$ then the lightest mass particle is on the order of experimental
values of proposed neutrino masses.