Bounds on the equation of state of cold nuclear matter from imaginary chemical potentials

Abstract

The sign problem in numerical calculations of the QCD Euclidean space path integral of QCD with a chemical potential vanishes if the chemical potential is imaginary and calculations of the partition function with an imaginary chemical potentials are equivalent to calculations with Lagrange multipliers enforcing the current density. At zero temperature, Lorentz boosts allows one to deduce properties of systems with both number density and and current density from properties of systems with a current density alone; this allows both upper and lower bounds to be determined for the equation of state (EOS) in the form of energy density as a function of number density.

Figures (1)

• Figure 1: Bounds from the classical $\phi^4$ theory discussed in the text. The left hand panel shows the ratio of the worst case upper bound (thin line) and optimal upper bound obtainable from Eq. (\ref{['Eq:ub']}) (thick line) energy density to the actual energy density as a function of density above unity and below unity shows the analogous results for the lower bound; the lower bound depends on $P/\epsilon$, so several values of $P/\epsilon$ are shown: zero (thick line), 1/6 (dashed line), 1/3 (dot-dashed line) and 1 (dotted line); the worst case is shown by a thin line which is almost overlapped by the dotted line of $P/\epsilon$=1, except at very low $n$. The left panel covers $n$ up to 15 since the energy is approaching its asymptotic value by then (as seen in the upper right hand panel). The thick line middle right hand panel show the actual values of $P/\epsilon$ which may help interpret the left-hand panel (with the thin line indicating $1/3$), while the bottom right panel indicates the extent to which a handful of measurements allows one to approximate the optimal bounds--focusing on the upper bound. In that figure, effects of measurements at $\frac{j_z g^2}{m^3} =.3, .6, 1.0 , 2.5, 5.0, 8.0$ and $13.0$ were simulated.