Renormalization group optimized $\lambda {\varphi }^4$ pressure at next-to-next-to-leading order

We investigate the renormalization group optimized perturbation theory (RGOPT) at the next-to-next-to-leading order (NNLO) for the thermal scalar field theory. From comparing three thus available successive RGOPT orders, we illustrate the efficient resummation and very good apparent convergence properties of the method. In particular, the remnant renormalization scale dependence of thermodynamical quantities is drastically improved as compared to both standard perturbative expansions and other related resummation methods, such as the screened perturbation theory. Our present results thus constitute a useful first NNLO illustration in view of NNLO applications of this approach to the more involved thermal QCD.

Figure 1

Free energy diagrams up to NNLO in $\lambda {\varphi }^4$ model.

Figure 2

The graphs being resummed at first nontrival RGOPT order.

Figure 3

LO (dotted lines) and NLO (dashed, green) RGOPT pressure $P/P_0(g\equiv \sqrt{\lambda /24})$ versus $\mathcal{O}(g^2)$ and $\mathcal{O}(g^{3/2})$ PT, and NLO SPT (for two different prescriptions, see main text). The different bands give the scale dependence for $\pi T\le \mu \le 4\pi T$.

Figure 4

Relative scale dependence at successive orders of the running coupling in the $\lambda {\varphi }^4$ model, given, respectively, at one-, two- and three-loop orders by Eqs. (3.8), (b8), and (b11), with $g\equiv \sqrt{\lambda /24}$ and $\pi T\le \mu \le 4\pi T$.

Figure 5

NNLO (three-loops) $\tilde{m}$ from the OPT prescription Eq. (3.2) (with $s_3=0$) and RG prescription Eqs. (3.3) and (a1) (with $s_3\ne 0$) as a function of $g\equiv \sqrt{\lambda /24}$, with the scale dependence $\pi T\le \mu \le 4\pi T$.

Figure 6

Comparison of NNLO pressure $P/P_{\text{ideal}}$ for the OPT and RG $\tilde{m}$ prescriptions (with $s_3=0$ and $s_3\ne 0$, respectively), as a function of $g\equiv \sqrt{\lambda /24}$ with $\pi T\le \mu \le 4\pi T$. NB: the lower pressure values correspond to the largest $\mu =4\pi T$ values due to infrared freedom of $\lambda {\varphi }^4$.

Figure 7

RGOPT pressure at successive LO,NLO,NNLO orders versus NLO,NNLO PT and NLO,NNLO SPT pressures, with ($g\equiv \sqrt{\lambda /24}$), $\pi T\le \mu \le 4\pi T$. NB: the lower pressure values correspond to the largest $\mu =4\pi T$ values due to infrared freedom of $\lambda {\varphi }^4$.

Figure 8

Top panel: OPT (dashed lines) and RG (thick lines) equations at NNLO as a function of $m$, for $s_3=0$, taking three-loop order running coupling $g(4\pi T)$ ($g\equiv \sqrt{\lambda /24}$) for two different values of the reference coupling, $g(2\pi T)=0.6$ and $g(2\pi T)=0.8$, respectively. Bottom panel: same captions for $s_3\ne 0$.