Thermodynamic formulation of vacuum energy density in flat spacetime and potential implications for the cosmological constant

TL;DR

The paper proposes a non-perturbative thermodynamic definition of vacuum energy density ${\rho_{\rm vac}}$ in flat spacetime across dimensions, extracting it from the high-temperature limit and linking Thermal and spatially compactified channels via modularity. In 2D integrable QFTs, ${\rho_{\rm vac}}$ is computed exactly via the Thermodynamic Bethe Ansatz with the Lightest Mass Bootstrap property, and explicit results are given for sine-Gordon, affine Toda, and related models, including phase-transition signatures. For free massive fields in $D=2,3,4$, the authors derive explicit expressions for ${\rho_{\rm vac}}$ in the Thermal channel and confirm modularity with the SpC channel, noting dimension-dependent signs and logarithmic terms in even dimensions. Applying the framework to 4D QCD through lattice data yields a negative estimator ${\rho_{\rm vac}} \approx - (200 \ \text{MeV})^{4}$, illustrating the potential relevance to cosmological constant considerations and highlighting the need for further cross-dimensional validation and connections to phenomenology.

Abstract

We propose a thermodynamical definition of the vacuum energy density $ρ_{\rm vac}$, defined as $\langle 0| T_{μν} |0\rangle = - ρ_{\rm vac} \, g_{μν}$, in quantum field theory in flat Minkowski space in $D$ spacetime dimensions, which can be computed in the limit of high temperature, namely in the limit $β= 1/T \to 0$. It takes the form $ρ_{\rm vac} = {\rm const} \cdot m^D$ where $m$ is a fundamental mass scale and ${\rm "const"}$ is a computable constant which can be positive or negative. Due to modular invariance $ρ_{\rm vac}$ can also be computed in a different non-thermodynamic channel where one spatial dimension is compactifed on a circle of circumference $β$ and we confirm this modularity for free massive theories for both bosons and fermions for $D=2,3,4$. We list various properties of $ρ_{\rm vac}$ that are generally required, for instance $ρ_{\rm vac}=0$ for conformal field theories, and others, such as the constraint that $ρ_{\rm vac}$ has opposite signs for free bosons verses fermions of the same mass, which is related to constraints from supersymmetry. Using the Thermodynamic Bethe Ansatz we compute $ρ_{\rm vac}$ exactly for 2 classes of integrable QFT's in $2D$ and interpreting some previously known results. We apply our definition of $ρ_{\rm vac}$ to Lattice QCD data with two light quarks (up and down) and one additional massive flavor (the strange quark), and find it is negative, $ρ_{\rm vac} \approx - ( 200 \, {\rm MeV} )^4$. Finally we make some remarks on the Cosmological Constant Problem since $ρ_{\rm vac}$ is central to any discussion of it.

Figures (5)

• Figure 1: An infinite cylinder in $D=d+1$ spacetime dimensions. For the thermal channel, $x$ is euclidean time and $y$ refers to $d$ spatial directions. For the SpC channel $x$ is a single compactified spatial coordinate and one $y$ coordinate is time $-\infty < t < \infty$.
• Figure 2: ${\rho_{\rm vac}}$ for the sine-Gordon model as a function of the coupling $\widehat{\beta}^2$.
• Figure 3: ${\rho_{\rm vac}}$ for the sinh-Gordon model (N=1) as a function of $b$ based on equation \ref{['gcalToda2']}.
• Figure 4: $p \beta^4$ as a function of temperature. (Taken from Bazavov).
• Figure 5: The pressure $p \,\beta^4$ as a function of $r=M\beta$ where $M = 1\,{\rm MeV}$. Data points are from Bazavov which refer to Figure \ref{['QCDfig']}, and the continuous curve is the fit \ref{['cQCDfit']}.