Non-perturbative Renormalization Research Articles

Nonperturbative renormalization group study of the stochastic Navier-Stokes equation

We study the renormalization group flow of the average action of the stochastic Navier-Stokes equation with power-law forcing. Using Galilean invariance, we introduce a nonperturbative approximation adapted to the zero-frequency sector of the theory in the parametric range of the Hölder exponent 4-2ε of the forcing where real-space local interactions are relevant. In any spatial dimension d, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for ε = 2 as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a saturation in the ε dependence of the scaling dimension of the eddy diffusivity at ε = 3/2 when, according to perturbative renormalization, the velocity field becomes infrared relevant.

The “tetrad only” theory space: nonperturbative renormalization flow and asymptotic safety

We set up a nonperturbative gravitational coarse graining flow and the corresponding functional renormalization group equation on the as to yet unexplored "tetrad only" theory space. It comprises action functionals which depend on the tetrad field (along with the related background and ghost fields) and are invariant under the semi-direct product of spacetime diffeomorphisms and local Lorentz transformations. This theory space differs from that of Quantum Einstein Gravity (QEG) in that the tetrad rather than the metric constitutes the fundamental variable and because of the additional symmetry requirement of local Lorentz invariance. It also differs from "Quantum Einstein Cartan Gravity" (QECG) investigated recently since the spin connection is not an independent field variable now. We explicitly compute the renormalization group flow on this theory space within the tetrad version of the Einstein-Hilbert truncation. A detailed comparison with analog results in QEG and QECG is performed in order to assess the impact the choice of a fundamental field variable has on the renormalization behavior of the gravitational average action, and the possibility of an asymptotically safe infinite cutoff limit is investigated. Implications for nonperturbative studies of fermionic matter coupled to quantum gravity are also discussed. It turns out that, in the context of functional flow equations, the "hybrid calculations" proposed in the literature (using the tetrad for fermionic diagrams only, and the metric in all others) are unlikely to be quantitatively reliable. Moreover we find that, unlike in perturbation theory, the non-propagating Faddeev-Popov ghosts related to the local Lorentz transformations may not be discarded but rather contribute quite significantly to the beta functions of Newton's constant and the cosmological constant.

Nonperturbative functional renormalization group for random field models and related disordered systems. IV. Supersymmetry and its spontaneous breaking

We apply the nonperturbative functional renormalization group (NP-FRG) in the superfield formalism that we have developed in the preceding paper to study long-standing issues concerning the critical behavior of the random field Ising model. Through the introduction of an appropriate regulator and a supersymmetry-compatible nonperturbative approximation, we are able to follow the supersymmetry, more specifically the superrotational invariance first unveiled by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)], and its spontaneous breaking along the RG flow. Breaking occurs below a critical dimension dDR \simeq 5.1, and the supersymmetry-broken fixed point that controls the critical behavior then leads to a breakdown of the "dimensional reduction" property. We solve the NP-FRG flow equations numerically and determine the critical exponents as a function of dimension down to d < 3, with a good agreement in d = 3 and d = 4 with the existing numerical estimates.

Nonperturbative functional renormalization group for random field models and related disordered systems. III. Superfield formalism and ground-state dominance

We reformulate the nonperturbative functional renormalization group for the random field Ising model in a superfield formalism, extending the supersymmetric description of the critical behavior of the system first proposed by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)]. We show that the two crucial ingredients for this extension are the introduction of a weighting factor, which accounts for ground-state dominance when multiple metastable states are present, and of multiple copies of the original system, which allows one to access the full functional dependence of the cumulants of the renormalized disorder and to describe rare events. We then derive exact renormalization group equations for the flow of the renormalized cumulants associated with the effective average action.

Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation

We present the implementation of the Blaizot-Méndez-Wschebor approximation scheme of the nonperturbative renormalization group we present in detail, which allows for the computation of the full-momentum dependence of correlation functions. We discuss its significance and its relation with other schemes, in particular, the derivative expansion. Quantitative results are presented for the test ground of scalar O(N) theories. Besides critical exponents, which are zero-momentum quantities, we compute the two-point function at criticality in the whole momentum range in three dimensions and, in the high-temperature phase, the universal structure factor. In all cases, we find very good agreement with the best existing results.

Lattice QCD calculation of nuclear parity violation

We present the first lattice QCD calculation of the leading-order momentum-independent parity violating coupling between pions and nucleons, $h_{\pi{NN}}^1$. The calculation performs measurements on dynamical anisotropic clover gauge configurations, with a spatial extent of $L\sim2.5$ fm, a spatial lattice spacing of $a_s\sim0.123$ fm, and a pion mass of $m_{\pi}\sim389$ MeV. While this first calculation does not include non-perturbative renormalization of the bare parity-violating operators, a chiral extrapolation to the physical pion mass, or contributions from disconnected (quark-loop) diagrams, these are expected to result in systematic errors within the quoted statistical error. We find a contribution from the `connected' diagrams of $h_{\pi{NN}}^{1,con}=(1.099\pm0.505^{+0.058}_{-0.064})\times10^{-7}$, consistent with current experimental bounds and previous model-dependent theoretical predictions.

Lattice quantum chromodynamics at the physical point and beyond

We review the work of the PACS-CS Collaboration, which aimed to realize lattice quantum chromodynamics (QCD) calculations at the physical point, i.e., those with quark masses set at physical values. This has been a long-term goal of lattice QCD simulation since its inception in 1979. After reviewing the algorithmic progress, which played a key role in this development, we summarize the simulations that explored the quark mass dependence of hadron masses down to values close to the physical point. In addition to allowing a reliable determination of the light hadron mass spectrum, this work provided clues on the validity range of chiral perturbation theory, which is widely used in phenomenology. We then describe the application of the technique of quark determinant reweighting, which enables lattice QCD calculations exactly on the physical point. The physical quark masses and the strong coupling constants are fundamental constants of the strong interaction. We describe a non-perturbative Schrödinger functional approach to figure out the non-perturbative renormalization needed to calculate them. There are a number of physical applications that can benefit from lattice QCD calculations carried out either near or at the physical point. We take up three illustrative examples: calculation of the physical properties of the ρ meson as a resonance, the electromagnetic form factor and charge radius of the pion, and charmed meson spectroscopy. Bringing single hadron properties under control opens up a number of new areas for serious lattice QCD research. One such area is electromagnetic effects in hadronic properties. We discuss the combined QCD plus QED simulation strategy and present results on electromagnetic mass difference. Another area is multi-hadron states, or nuclei. We discuss the motivations and difficulties in this area, and describe our work for deuteron and helium as our initial playground. We conclude with a brief discussion on the future perspective of lattice QCD.

Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation: General framework and first applications

We present an analytical method, rooted in the nonperturbative renormalization group, that allows one to calculate the critical exponents and the correlation and response functions of the Kardar-Parisi-Zhang (KPZ) growth equation in all its different regimes, including the strong-coupling one. We analyze the symmetries of the KPZ problem and derive an approximation scheme that satisfies the linearly realized ones. We implement this scheme at the minimal order in the response field, and show that it yields a complete, qualitatively correct phase diagram in all dimensions, with reasonable values for the critical exponents in physical dimensions. We also compute in one dimension the full (momentum and frequency dependent) correlation function, and the associated universal scaling function. We find a very satisfactory quantitative agreement with the exact result from Prähofer and Spohn [J. Stat. Phys. 115, 255 (2004)]. In particular, we obtain for the universal amplitude ratio g_{0}≃1.149(18), to be compared with the exact value g_{0}=1.1504... (the Baik and Rain [J. Stat. Phys. 100, 523 (2000)] constant). We emphasize that all these results, which can be systematically improved, are obtained with sole input the bare action and its symmetries, without further assumptions on the existence of scaling or on the form of the scaling function.

Link between the hierarchical reference theory of liquids and a new version of the non-perturbative renormalization group in statistical field theory

I propose a new derivation of the Hierarchical Reference Theory of liquids. Two formalisms, one in the grand canonical ensemble, the other in the framework of statistical field theory are given in parallel. In the latter the theory is an avatar of a new version of the non-perturbative renormalization group [J.-M. Caillol, J. Phys. A : Math. Gen. 42, 225004 (2009)]. The flow of the Wilsonian action as well as that of the effective average action of Wetterich are derived and a simple relation between the two functionals is established. The standard Hierarchical Reference Theory for liquids [A. Parola and L. Reatto, Adv. Phys. 44, 211 (1995)] is recovered for a sharp infra-red cut-off of the propagator.

Nonperturbative renormalization group approach to strongly correlated lattice bosons

We present a nonperturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the renormalization-group flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with quantum Monte Carlo simulations, and reproduces the two universality classes of the superfluid--Mott-insulator transition. The critical behavior near the multicritical points, where the transition takes place at constant density, agrees with the original predictions of Fisher {\it et al.} [Phys. Rev. B {\bf 40}, 546 (1989)] based on simple scaling arguments. At a generic transition point, the critical behavior is mean-field like with logarithmic corrections in two dimensions. In the weakly-correlated superfluid phase (far away from the Mott insulating phase), the renormalization-group flow is controlled by the Bogoliubov fixed point down to a characteristic (Ginzburg) momentum scale $k_G$ which is much smaller than the inverse healing length $k_h$. In the vicinity of the multicritical points, when the density is commensurate, we identify a sharp crossover from a weakly- to a strongly-correlated superfluid phase where the condensate density and the superfluid stiffness are strongly suppressed and both $k_G$ and $k_h$ are of the order of the inverse lattice spacing.

Hydrodynamic collective modes for cold trapped gases

We suggest that collective oscillation frequencies of cold trapped gases can be used to test predictions from quantum many-body physics. Our motivation lies in both rigid experimental tests of theoretical calculations and a possible improvement of measurements of particle number, chemical potential or temperature. We calculate the effects of interaction, dimensionality and thermal fluctuations on the collective modes of a dilute Bose gas in the hydrodynamic limit. The underlying equation of state is provided by the non-perturbative Functional Renormalization Group or by Lee–Yang theory. The spectrum of oscillation frequencies could be measured by response techniques. Our findings are generalized to bosonic or fermionic quantum gases with an arbitrary equation of state in the two-fluid hydrodynamic regime. For any given equation of state P(μ, T) and normal fluid density nn(μ, T), the collective oscillation frequencies in a d-dimensional isotropic potential are found to be the eigenvalues of an ordinary differential operator. We suggest a method of numerical solution and discuss the zero-temperature limit. Exact results are provided for harmonic traps and certain special forms of the equation of state. We also present a phenomenological treatment of dissipation effects and discuss the possibility of exciting the different eigenmodes individually.

General framework of the non-perturbative renormalization group for non-equilibrium steady states

This paper is devoted to presenting in detail the non-perturbative renormalization group (NPRG) formalism to investigate out-of-equilibrium systems and critical dynamics in statistical physics. The general NPRG framework for studying non-equilibrium steady states in stochastic models is expounded and fundamental technicalities are stressed, mainly regarding the role of causality and of It’s discretization. We analyze the consequences of It’s prescription in the NPRG framework and eventually provide an adequate regularization to encode them automatically. Besides, we show how to build a supersymmetric NPRG formalism with emphasis on time-reversal symmetric problems, whose supersymmetric structure allows for a particularly simple implementation of NPRG in which causality issues are transparent. We illustrate the two approaches on the example of Model A within the derivative expansion approximation at order 2 and check that they yield identical results. We stress, though, that the framework presented here also applies to genuinely out-of-equilibrium problems.

The non-perturbative renormalization group in the ordered phase

We study some analytical properties of the solutions of the non-perturbative renormalization group flow equations for a scalar field theory with Z 2 symmetry in the ordered phase, i.e. at temperatures below the critical temperature. The study is made in the framework of the local potential approximation. We show that the required physical discontinuity of the magnetic susceptibility χ ( M ) at M = ± M 0 ( M 0 spontaneous magnetization) is reproduced only if the cut-off function which separates high and low energy modes satisfies to some restrictive explicit mathematical conditions; we stress that these conditions are not satisfied by a sharp cut-off in dimensions of space d < 4 . By generalizing a method proposed earlier by Bonanno and Lacagnina [Nucl. Phys. B 693 (2004) 36] to any kind of cut-off we propose to solve numerically the renormalization group flow equations for the threshold functions rather than for the local potential. It yields an algorithm sufficiently robust and precise to extract universal as well as non-universal quantities from numerical experiments at any temperature, in particular at sub-critical temperatures in the ordered phase. Numerical results obtained for the φ 4 potential with three different cut-off functions are reported and compared. The data confirm our theoretical predictions concerning the analytical behavior of χ ( M ) at M = ± M 0 . Fixed point solutions of the adimensioned renormalization group flow equations are also obtained in the same vein, that is by solving the fixed points equations and the associated eigenvalue problem for the threshold functions rather than for the potential. We report high precision data for the odd and even spectra of critical exponents for different cut-offs obtained in this way.

Precision computation of the kaon bag parameter

Indirect CP violation in K→ππ decays plays a central role in constraining the flavor structure of the Standard Model (SM) and in the search for new physics. For many years the leading uncertainty in the SM prediction of this phenomenon was the one associated with the nonperturbative strong interaction dynamics in this process. Here we present a fully controlled lattice QCD calculation of these effects, which are described by the neutral kaon mixing parameter BK. We use a two step HEX smeared clover-improved Wilson action, with four lattice spacings from a≈0.054 fm to a≈0.093 fm and pion masses at and even below the physical value. Nonperturbative renormalization is performed in the RI-MOM scheme, where we find that operator mixing induced by chiral symmetry breaking is very small. Using fully nonperturbative continuum running, we obtain our main result BKRI(3.5 GeV)=0.531(6)stat(2)sys. A perturbative 2-loop conversion yields BKMS¯-NDR(2 GeV)=0.564(6)stat(3)sys(6)PT and BˆK=0.773(8)stat(3)sys(8)PT, which is in good agreement with current results from fits to experimental data.

Beyond Miransky scaling

We study the scaling behavior of physical observables in strongly-flavored asymptotically free gauge theories, such as many-flavor QCD. Such theories approach a quantum critical point when the number of fermion flavors is increased. It is well-known that physical observables at this quantum critical point exhibit an exponential scaling behavior (Miransky scaling), provided the gauge coupling is considered as a constant external parameter. This scaling behavior is modified when the scale dependence of the gauge coupling is taken into account. Provided that the gauge coupling approaches an IR fixed point, we derive the resulting universal power-law corrections to the exponential scaling behavior and show that they are uniquely determined by the IR critical exponent of the gauge coupling. To illustrate our findings, we compute the universal corrections in many-flavor QCD with the aid of nonperturbative functional renormalization group methods. In this case, we expect the power-law scaling to be quantitatively more relevant if the theories are probed, for instance, at integer ${N}_{\mathrm{f}}$ as done in lattice simulations.

Continuum limit ofBKfrom2+1flavor domain wall QCD

We determine the neutral kaon mixing matrix element ${B}_{K}$ in the continuum limit with $2+1$ flavors of domain wall fermions, using the Iwasaki gauge action at two different lattice spacings. These lattice fermions have near exact chiral symmetry and therefore avoid artificial lattice operator mixing. We introduce a significant improvement to the conventional nonperturbative renormalization (NPR) method in which the bare matrix elements are renormalized nonperturbatively in the regularization invariant momentum scheme (RI-MOM) and are then converted into the $\overline{\mathrm{MS}}$ scheme using continuum perturbation theory. In addition to RI-MOM, we introduce and implement four nonexceptional intermediate momentum schemes that suppress infrared nonperturbative uncertainties in the renormalization procedure. We compute the conversion factors relating the matrix elements in this family of regularization invariant symmetric momentum schemes (RI-SMOM) and $\overline{\mathrm{MS}}$ at one-loop order. Comparison of the results obtained using these different intermediate schemes allows for a more reliable estimate of the unknown higher-order contributions and hence for a correspondingly more robust estimate of the systematic error. We also apply a recently proposed approach in which twisted boundary conditions are used to control the Symanzik expansion for off-shell vertex functions leading to a better control of the renormalization in the continuum limit. We control chiral extrapolation errors by considering both the next-to-leading order SU(2) chiral effective theory, and an analytic mass expansion. We obtain ${B}_{K}^{\overline{\mathrm{MS}}}(3\text{ }\text{ }\mathrm{GeV})=0.529(5{)}_{\mathrm{stat}}(15{)}_{\ensuremath{\chi}}(2{)}_{\mathrm{FV}}(11{)}_{\mathrm{NPR}}$. This corresponds to ${\stackrel{^}{B}}_{K}^{\overline{\mathrm{RGI}}}=0.749(7{)}_{\mathrm{stat}}(21{)}_{\ensuremath{\chi}}(3{)}_{\mathrm{FV}}(15{)}_{\mathrm{NPR}}$. Adding all sources of error in quadrature, we obtain ${\stackrel{^}{B}}_{K}^{\overline{\mathrm{RGI}}}=0.749(27{)}_{\mathrm{combined}}$, with an overall combined error of 3.6%.

Matching factors forΔS=1four-quark operators in RI/SMOM schemes

The nonperturbative renormalization of four-quark operators plays a significant role in lattice studies of flavor physics. For this purpose, we define regularization-independent symmetric momentum-subtraction (RI/SMOM) schemes for $\ensuremath{\Delta}S=1$ flavor-changing four-quark operators and provide one-loop matching factors to the $\overline{\mathrm{MS}}$ scheme in naive dimensional regularization. The mixing of two-quark operators is discussed in terms of two different classes of schemes. We provide a compact expression for the finite one-loop amplitudes, which allows for a straightforward definition of further RI/SMOM schemes.

Infrared behavior of interacting bosons at zero temperature

We review the infrared behavior of interacting bosons at zero temperature. After a brief discussion of the Bogoliubov approximation and the breakdown of perturbation theory due to infrared divergences, we present two approaches that are free of infrared divergences -- Popov's hydrodynamic theory and the non-perturbative renormalization group -- and allow us to obtain the exact infrared behavior of the correlation functions. We also point out the connection between the infrared behavior in the superfluid phase and the critical behavior at the superfluid--Mott-insulator transition in the Bose-Hubbard model.

INFRARED BEHAVIOR OF INTERACTING BOSONS AT ZERO TEMPERATURE

We review the infrared behavior of interacting bosons at zero temperature. After a brief discussion of the Bogoliubov approximation and the breakdown of perturbation theory due to infrared divergences, we show how the non-perturbative renormalization group enables to obtain the exact infrared behavior of the correlation functions.

NON-PERTURBATIVE RENORMALIZATION GROUP: BASIC PRINCIPLES AND SOME APPLICATIONS

The non-perturbative renormalization group (NPRG), in its modern form, constitutes an efficient framework to investigate the physics of systems whose long-distance behavior is dominated by strong fluctuations that are out of reach of perturbative approaches. We present here the basic principles underlying the NPRG and illustrate its power in the context of two longstanding problems of condensed matter and soft matter physics: the nature of the phase transition occuring in frustrated magnets in three dimensions and the phase diagram of polymerized phantom membranes.