Spectral representation of the shear viscosity for local scalar QFTs at finite temperature

In local scalar quantum field theories at finite temperature correlation functions are known to satisfy certain nonperturbative constraints, which for two-point functions in particular implies the existence of a generalization of the standard Källén-Lehmann representation. In this work, we use these constraints in order to derive a spectral representation for the shear viscosity arising from the thermal asymptotic states, ${\eta }_0$. As an example, we calculate ${\eta }_0$ in ${\varphi }^4$ theory, establishing its leading behavior in the small and large coupling regimes.

Figure 1

${\rho }_{\pi \pi }(p_0)/T^4$ plotted as a function of $p_0/T$ for fixed $m/T$ (dashed lines) and fixed $|\lambda |$ (solid lines).

Figure 2

${\eta }_0/T^3$ plotted as a function of $|\lambda |$ for varying $m/T$.

Figure 3

$\frac{1}{20T^3}\frac{d{\rho }_{\pi \pi }}{d{p}_0}$ plotted as a function of $p_0/T$ and $|\lambda |$ for $m/T=0.1$. The blue and red lines correspond to the fixed surfaces $\frac{p_0}{T}=0$ and $\lambda =0$, respectively.