Extended Renormalization Group Research Articles

All-coupling theory for the Fröhlich polaron

The Fr\"ohlich model describes the interaction of a mobile impurity with a surrounding bath of phonons which leads to the formation of a quasiparticle, the polaron. In this article an efficient renormalization group approach is presented which provides a description of Fr\"ohlich polarons in all regimes ranging from weak- to strong coupling. We apply the method to the Bose polaron problem of an ultracold impurity atom interacting with a background gas that is Bose-condensed. The extended renormalization group approach introduced here is capable to predict ground state properties for arbitrarily small impurity masses. This allows us to obtain the full phase diagram of the corresponding Bogoliubov-Fr\"ohlich Hamiltonian, characterized by two dimensionless coupling constants. Our method is benchmarked by comparison of the ground state energy to recent diagrammatic quantum Monte Carlo calculations.

Erratum: Scale setting using the extended renormalization group and the principle of maximum conformality: The QCD coupling constant at four loops [Phys. Rev. D85, 034038 (2012)

A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The extended renormalization group equations, which express the invariance of physical observables under both the renormalization scale- and scheme-parameter transformations, provide a convenient way for estimating the scale- and scheme-dependence of the physical process. In this paper, we present a solution for the scale-equation of the extended renormalization group equations at the four-loop level. Using the principle of maximum conformality (PMC) / Brodsky-Lepage-Mackenzie (BLM) scale-setting method, all non-conformal $\{\beta_i\}$ terms in the perturbative expansion series can be summed into the running coupling, and the resulting scale-fixed predictions are independent of the renormalization scheme. Different schemes lead to different effective PMC/BLM scales, but the final results are scheme independent. Conversely, from the requirement of scheme independence, one not only can obtain scheme-independent commensurate scale relations among different observables, but also determine the scale displacements among the PMC/BLM scales which are derived under different schemes. In principle, the PMC/BLM scales can be fixed order-by-order, and as a useful reference, we present a systematic and scheme-independent procedure for setting PMC/BLM scales up to NNLO. An explicit application for determining the scale setting of $R_{e^{+}e^-}(Q)$ up to four loops is presented. By using the world average $\alpha^{\bar{MS}}_s(M_Z) =0.1184 \pm 0.0007$, we obtain the asymptotic scale for the 't Hooft associated with the $\bar{MS}$ scheme, $\Lambda^{'tH}_{\bar{MS}}= 245^{+9}_{-10}$ MeV, and the asymptotic scale for the conventional $\bar{MS}$ scheme, $\Lambda_{\bar{MS}}= 213^{+19}_{-8}$ MeV.

Scale setting using the extended renormalization group and the principle of maximum conformality: The QCD coupling constant at four loops

A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The extended renormalization group equations, which express the invariance of the physical observables under both the renormalization scale- and scheme-parameter transformations, provide a convenient way for estimating the scale- and scheme-dependence of the physical process. In this paper, we present a solution for the scale equation of the extended renormalization group equations at the four-loop level. Using the principle of maximum conformality (PMC)/Brodsky-Lepage-Mackenzie (BLM) scale-setting method, all nonconformal ${{\ensuremath{\beta}}_{i}}$ terms in the perturbative expansion series can be summed into the running coupling, and the resulting scale-fixed predictions are independent of the renormalization scheme. The PMC/BLM scales can be fixed order-by-order. As a useful reference, we present a systematic and scheme-independent procedure for setting PMC/BLM scales up to next-to-next-to-leading order. An explicit application for determining the scale setting of ${R}_{{e}^{+}{e}^{\ensuremath{-}}}(Q)$ up to four loops is presented. By using the world average ${\ensuremath{\alpha}}_{s}^{\overline{\mathrm{MS}}}({M}_{Z})=0.1184\ifmmode\pm\else\textpm\fi{}0.0007$, we obtain the asymptotic scale for the 't Hooft scheme associated with the $\overline{\mathrm{MS}}$ scheme, ${\ensuremath{\Lambda}}_{\overline{\mathrm{MS}}}^{\ensuremath{'}tH}={245}_{\ensuremath{-}10}^{+9}\text{ }\text{ }\mathrm{MeV}$, and the asymptotic scale for the conventional $\overline{\mathrm{MS}}$ scheme, ${\ensuremath{\Lambda}}_{\overline{\mathrm{MS}}}={213}_{\ensuremath{-}8}^{+19}\text{ }\text{ }\mathrm{MeV}$.

Nonrelativistic inverse square potential, scale anomaly, and complex extension

The old problem of a singular, inverse square potential in nonrelativistic quantum mechanics is treated employing a field-theoretic, functional renormalization method. An emergent contact coupling flows to a fixed point or develops a limit cycle depending on the discriminant of its quadratic beta function. We analyze the fixed points in both conformal and non-conformal phases and perform a natural extension of the renormalization group analysis to complex values of the contact coupling. Physical interpretation and motivation for this extension is the presence of an inelastic scattering channel in two-body collisions. We present a geometric description of the complex generalization by considering renormalization group flows on the Riemann sphere. Finally, using bosonization, we find an analytical solution of the extended renormalization group flow equations, constituting the main result of our work.

Characteristics of round thermal discharging in a flowing environment

The 3-D round thermal buoyant jets discharged from a submerged port in a flowing environment were investigated using the buoyancy extended renormalization group (RNG) based on the k–ɛ model, which was numerically solved by the finite analytic method (FAM). Laboratory experiments were conducted at different cross-to-jet velocity ratios to complement and verify the theoretical analysis with the numerical model. The simulated physical parameters such as configuration of the flow, the bifurcation phenomena, and the temperature and velocity trajectories were generally in good qualitative agreement with the experimental observation. All trends were correctly predicted: the trajectories based on maximum velocity and maximum temperature are different, the included angle of the two high temperature regions in the cross -section is proved to be in the same region as that in the air, and the most characteristic parameter for 3-D round buoyant jets in a flowing environment is the jet-to-crossflow velocity ratio. It is suggested that the buoyancy extended RNG based on the k–ɛ model combined with the FAM can be used to simulate the 3-D round heated buoyant jets discharged from a submerged port in a flowing environment.

Non-Ladder Extended Renormalization Group Analysis of the Dynamical Chiral Symmetry Breaking

The order parameters of dynamical chiral symmetry breaking in QCD, the dynamical mass of quarks and the chiral condensates, are evaluated by numerically solving the Non-Perturbative Renormalization Group (NPRG) equations. We employ an approximation scheme beyond ``the ladder'', that is, beyond the (improved) ladder Schwinger-Dyson equations. The chiral condensates are enhanced compared with the ladder ones, which is phenomenologically favorable. The gauge dependence of the order parameters is fairly reduced in this scheme.

Lennard-Jones as a model for argon and test of extended renormalization group calculations

Renormalization group (RG) procedures have been extended recently in phase-space cell approximation to predict, in addition to universal thermal properties observed asymptotically close to the gas-liquid critical point of fluids, also nonuniversal and nonasymptotic properties. This “globalized” RG theory is applied here, using a Lennard-Jones potential, to calculate the temperature, density, and pressure at the critical point of argon and to calculate pressures for a wide range of densities at temperatures close to, below, and considerably above that at the argon critical point. Choices required for the Lennard-Jones parameters and the quality of fit to experimental data suggest some of the strengths and limitations of the global RG theory.

Relating physical observables in QCD using the extended renormalization group method.

We discuss the St\"uckelberg-Peterman extended renormalization group equations in perturbative QCD, which express the invariance of physical observables under renormalization-scale and scheme-parameter transformations. We introduce a universal coupling function that covers all possible choices of scale and scheme. Any perturbative series in QCD is shown to be equivalent to a particular point in this function. This function can be computed from a set of first-order differential equations involving the extended beta functions. We propose the use of these evolution equations instead of perturbative series for numerical evaluation of physical observables. This formalism is free of scale-scheme ambiguity and allows a reliable error analysis of higher-order corrections. It also provides a precise definition for $\Lambda_{\overline{\rm MS}}$ as the pole in the associated 't Hooft scheme. A concrete application to $R(e^+e^- \to {\rm hadrons})$ is presented.

A complete proof of the Feigenbaum conjectures

The Feigenbaum phenomenon is studied by analyzing an extended renormalization group map ℳ. This map acts on functionsΦ that are jointly analytic in a “position variable” (t) and in the parameter (μ) that controls the period doubling phenomenon. A fixed pointΦ * for this map is found. The usual renormalization group doubling operatorN acts on this functionΦ * simply by multiplication ofμ with the universal Feigenbaum ratioδ *= 4.669201..., i.e., (N Φ *(μ,t)=Φ *(δ * μ,t). Therefore, the one-parameter family of functions,Ψ * ,Ψ * (t)=(Φ *(μ,t), is invariant underN. In particular, the functionΨ 0 * is the Feigenbaum fixed point ofN, whileΨ * represents the unstable manifold ofN. It is proven that this unstable manifold crosses the manifold of functions with superstable period two transversally.