Constraints on a special running vacuum model
  We study a special running vacuum model (RVM) with $\Lambda = 3 \alpha
H^2+3\beta H_0^4 H^{-2}+\Lambda_0$, where $\alpha$, $\beta$ and $\Lambda_0$ are
the model parameters and $H$ is the Hubble one. This RVM has non-analytic
background solutions for the energy densities of matter and radiation, which
can only be evaluated numerically. From the analysis of the CMB power spectrum
and baryon acoustic oscillation along with the prior of $\alpha>0$ to avoid
having a negative dark energy density, we find that $\alpha<2.83\times 10^{-4}$
and $\beta=(-0.2^{+3.9}_{-4.5})\times 10^{-4}$ (95$\%$ C.L.). We show that the
RVM fits the cosmological data comparably to the $\Lambda$CDM. In addition, we
relate the fluctuation amplitude $\sigma_8$ to the neutrino mass sum $\Sigma
m_\nu$.
