General Renormalization Group Research Articles

Functional flows for complex effective actions

In the present work we set up a general functional renormalisation group framework for the computation of complex effective actions. For explicit computations we consider both flows of the Wilsonian effective action and the one-particle irreducible (1PI) effective action. The latter is based on an appropriate definition of a Legendre transform for complex actions, and we show its validity by comparison to exact results in zero dimensions, as well as a comparison to results for the Wilsonian effective action. In the present implementations of the general approaches, the flow of the Wilsonian effective action has a wider range of applicability and we obtain results for the effective potential of complex fields in \phi^4ϕ4-theories from zero up to four dimensions. These results are also compared with results from the 1PI effective action within its range of applicability. The complex effective action also allows us to determine the location of the Lee-Yang zeros for general parameter values. We also discuss the extension of the present results to general theories including QCD.

Towards general scalar-Yukawa renormalisation group equations at three-loop order

For arbitrary four-dimensional quantum field theories with scalars and fermions, renormalisation group equations in the overline{mathrm{MS}} scheme are investigated at three-loop order in perturbation theory. Collecting literature results, general expressions are obtained for field anomalous dimensions, Yukawa interactions, as well as fermion masses. The renormalisation group evolution of scalar quartic, cubic and mass terms is determined up to a few unknown coefficients. The combined results are applied to compute the renormalisation group evolution of the gaugeless Litim-Sannino model.

Frustrated quantum spins at finite temperature: Pseudo-Majorana functional renormalization group approach

The pseudofermion functional renormalization group (PFFRG) method has proven to be a powerful numerical approach to treat frustrated quantum spin systems. In its usual implementation, however, the complex fermionic representation of spin operators introduces unphysical Hilbert-space sectors which render an application at finite temperatures inaccurate. In this work we formulate a general functional renormalization group approach based on Majorana fermions to overcome these difficulties. We, particularly, implement spin operators via an $\text{SO}(3)$ symmetric Majorana representation which does not introduce any unphysical states and, hence, remains applicable to quantum spin models at finite temperatures. We apply this scheme, dubbed pseudo-Majorana functional renormalization group (PMFRG) method, to frustrated Heisenberg models on small spin clusters as well as square and triangular lattices. Computing the finite-temperature behavior of spin correlations and thermodynamic quantities such as free energy and heat capacity, we find good agreement with exact diagonalization and the high-temperature series expansion down to moderate temperatures. We observe a significantly enhanced accuracy of the PMFRG compared to the PFFRG at finite temperatures. More generally, we conclude that the development of functional renormalization group approaches with Majorana fermions considerably extends the scope of applicability of such methods.

Lessons from conformally reduced quantum gravity

In this work we study a significantly enlarged truncation of conformally reduced quantum gravity in the context of asymptotic safety, including all operators that can be resolved in such a truncation including up to the sixth order in derivatives. A fixed point analysis suggests that there is no asymptotically safe fixed point in this system once one goes beyond an Einstein–Hilbert approximation. We will put these findings into context and discuss some lessons that can be learned from these results for general non-perturbative renormalisation group flows.

General scalar renormalisation group equations at three-loop order

For arbitrary scalar QFTs in four dimensions, renormalisation group equations of quartic and cubic interactions, mass terms, as well as field anomalous dimensions are computed at three-loop order in the overline{mathrm{MS}} scheme. Utilising pre-existing literature expressions for a specific model, loop integrals are avoided and templates for general theories are obtained. We reiterate known four-loop expressions, and from those derive β functions for scalar masses and cubic interactions. As an example, the results are applied to compute all renormalisation group equations in U(n) × U(n) scalar theories.

Running vacuum in quantum field theory in curved spacetime: renormalizing rho _{vac} without sim m^4 terms

The Lambda -term in Einstein’s equations is a fundamental building block of the ‘concordance’ Lambda CDM model of cosmology. Even though the model is not free of fundamental problems, they have not been circumvented by any alternative dark energy proposal either. Here we stick to the Lambda -term, but we contend that it can be a ‘running quantity’ in quantum field theory (QFT) in curved space time. A plethora of phenomenological works have shown that this option can be highly competitive with the Lambda CDM with a rigid cosmological term. The, so-called, ‘running vacuum models’ (RVM’s) are characterized by the vacuum energy density, rho _{vac}, being a series of (even) powers of the Hubble parameter and its time derivatives. Such theoretical form has been motivated by general renormalization group arguments, which look plausible. Here we dwell further upon the origin of the RVM structure within QFT in FLRW spacetime. We compute the renormalized energy-momentum tensor with the help of the adiabatic regularization procedure and find that it leads essentially to the RVM form. This means that rho _{vac}(H) evolves as a constant term plus dynamical components {{mathcal {O}}}(H^2) and mathcal{O}(H^4), the latter being relevant for the early universe only. However, the renormalized rho _{vac}(H) does not carry dangerous terms proportional to the quartic power of the masses (sim m^4) of the fields, these terms being a well-known source of exceedingly large contributions. At present, rho _{vac}(H) is dominated by the additive constant term accompanied by a mild dynamical component sim nu H^2 (|nu |ll 1), which mimics quintessence.

Exact β-Function in Abelian and non-Abelian $${\cal N} = 1$$ Supersymmetric Gauge Models and Its Analogy with the QCD β-Function in the C-scheme

For $${\cal N} = 1$$ supersymmetric Yang—Mills theory without matter it is demonstrated that there is a class of renormalization schemes, in which the exact Novikov, Shifman, Vainshtein, and Zakharov (NSVZ) formula for the renormalization group β-function, defined in terms of the renormalized coupling constant, is valid. These schemes are related with each other by finite renormalizations forming a one-parameter commutative subgroup of general renormalization group transformations. The analogy between the exact β-function in $${\cal N} = 1$$ supersymmetric Yang-Mills theory without matter and the β-function of quantum chromodynamics in the C-scheme is discussed.

On Renormalizations in Nonrenormalizable Theories

A new view of the procedure of renormalizations in nonrenormalizable theories is proposed. This view is based on the standard procedure of the BPHZ R-operation, which is equally applicable to any local quantum field theory irrespective of renormalizability. The key point is that the multiplicative renormalization used in renormalizable theories is replaced by an operation in which the renormalization constant depends on the momenta over which integration in subgraphs is performed. In this case, the requirement for the counterterms to be local (precisely as in renormalizable theories) leads to recurrence relations between leading, subleading, etc., ultraviolet divergences in all orders of perturbation theory. This allows one to obtain generalized renormalization group equations for scattering amplitudes, which have an integro-differential form and lead to the summation of the leading asymptotics, just as in renormalizable theories.

\u21242 breaking effects in 2-loop RG evolution of 2HDM

We investigate the effects of a ℤ2 symmetry in the mathcal{C}mathcal{P} -conserving Two-Higgs-Doublet-Model (2HDM); which is often imposed to prevent Flavor-Changing-Neutral-Currents (FCNCs) at tree-level. Specifically, we analyze how a breaking of the ℤ2 symmetry spreads during renormalization group evolution; employing general 2-loop renormalization group equations that we have derived. Evolving the model from the electroweak to the Planck scale, we find that while the case of an exact ℤ2 symmetric 2HDM is very constrained, a soft breaking of the ℤ2 symmetry extends the valid parameter space regions. The effects of a hard ℤ2 breaking in the scalar sector as well as the stability of the flavor alignment ansatz are also investigated. We find that while a hard breaking of the ℤ2 symmetry in the potential is problematic, since it speeds up the growth of quartic couplings, the generated FCNCs are heavily suppressed. Conversely, we also find that hard ℤ2 breaking in the Yukawa sector at most gives moderate ℤ2 breaking in the potential; whereas the FCNCs can become quite sizable far away from the ℤ2 symmetric regions.

Phases Of Adjoint QCD$_3$ And Dualities

We study 2+1 dimensional gauge theories with a Chern-Simons term and a fermion in the adjoint representation. We apply general considerations of symmetries, anomalies, and renormalization group flows to determine the possible phases of the theory as a function of the gauge group, the Chern-Simons level kk, and the fermion mass. We propose an inherently quantum mechanical phase of adjoint QCD with small enough kk, where the infrared is described by a certain Topological Quantum Field Theory (TQFT). For a special choice of the mass, the theory has {\cal N}=1 supersymmetry. There this TQFT is accompanied by a massless Majorana fermion – a Goldstino signaling spontaneous supersymmetry breaking. Our analysis leads us to conjecture a number of new infrared fermion-fermion dualities involving SUSU, SOSO, and SpSp gauge theories. It also leads us to suggest a phase diagram of SO/SpSO/Sp gauge theories with a fermion in the traceless symmetric/antisymmetric tensor representation of the gauge group.

Structure of UV divergences in maximally supersymmetric gauge theories

We consider the UV divergences up to sub-subleading order for the four-point on-shell scattering amplitudes in $D=8$ supersymmetric Yang-Mills theory in the planar limit. We trace how the leading, subleading, etc divergences appear in all orders of perturbation theory. The structure of these divergences is typical for any local quantum field theory independently on renormalizability. We show how the generalized renormalization group equations allow one to evaluate the leading, subleading, etc. contributions in all orders of perturbation theory starting from one-, two-, etc. loop diagrams respectively. We focus then on subtraction scheme dependence of the results and show that in full analogy with renormalizable theories the scheme dependence can be absorbed into the redefinition of the couplings. The only difference is that the role of the couplings play dimensionless combinations like ${g}^{2}{s}^{2}$ or ${g}^{2}{t}^{2}$, where $s$ and $t$ are the Mandelstam variables.

Superuniversality of topological quantum phase transition and global phase diagram of dirty topological systems in three dimensions

The quantum phase transition between two clean, non interacting topologically distinct gapped states in three dimensions is governed by a massless Dirac fermion fixed point, irrespective of the underlying symmetry class, and this constitutes a remarkably simple example of superuniversality. For a sufficiently weak disorder strength, we show that the massless Dirac fixed point is at the heart of the robustness of superuniversality. We establish this by considering both perturbative and nonperturbative effects of disorder. The superuniversality breaks down at a critical strength of disorder, beyond which the topologically distinct localized phases become separated by a delocalized diffusive phase. In the global phase diagram, the disorder controlled fixed point where superuniversality is lost, serves as a multicritical point, where the delocalized diffusive and two topologically distinct localized phases meet and the nature of the localization-delocalization transition depends on the underlying symmetry class. Based on these features we construct the global phase diagrams of noninteracting, dirty topological systems in three dimensions. We also establish a similar structure of the phase diagram and the superuniversality for weak disorder in higher spatial dimensions. By noting that $1/r^2$ power-law correlated disorder acts as a marginal perturbation for massless Dirac fermion in any spatial dimension $d$, we have established a general renormalization group framework for addressing disorder driven critical phenomena for fixed spatial dimension $d > 2$.

Nonequilibrium dynamical renormalization group: Dynamical crossover from weak to infinite randomness in the transverse-field Ising chain

In this work we formulate the nonequilibrium dynamical renormalization group (ndRG). The ndRG represents a general renormalization-group scheme for the analytical description of the real-time dynamics of complex quantum many-body systems. In particular, the ndRG incorporates time as an additional scale which turns out to be important for the description of the long-time dynamics. It can be applied to both translational-invariant and disordered systems. As a concrete application, we study the real-time dynamics after a quench between two quantum critical points of different universality classes. We achieve this by switching on weak disorder in a one-dimensional transverse-field Ising model initially prepared at its clean quantum critical point. By comparing to numerically exact simulations for large systems, we show that the ndRG is capable of analytically capturing the full crossover from weak to infinite randomness. We analytically study signatures of localization in both real space and Fock space.

Fickian dispersion is anomalous

The thesis put forward here is that the occurrence of Fickian dispersion in geophysical settings is a rare event and consequently should be labeled as anomalous. What people classically call anomalous is really the norm. In a Lagrangian setting, a process with mean square displacement which is proportional to time is generally labeled as Fickian dispersion. With a number of counter examples we show why this definition is fraught with difficulty. In a related discussion, we show an infinite second moment does not necessarily imply the process is super dispersive. By employing a rigorous mathematical definition of Fickian dispersion we illustrate why it is so hard to find a Fickian process. We go on to employ a number of renormalization group approaches to classify non-Fickian dispersive behavior. Scaling laws for the probability density function for a dispersive process, the distribution for the first passage times, the mean first passage time, and the finite-size Lyapunov exponent are presented for fixed points of both deterministic and stochastic renormalization group operators. The fixed points of the renormalization group operators are p-self-similar processes. A generalized renormalization group operator is introduced whose fixed points form a set of generalized self-similar processes. Power-law clocks are introduced to examine multi-scaling behavior. Several examples of these ideas are presented and discussed.

Logarithmic Correction for the Susceptibility of the 4-Dimensional Weakly Self-Avoiding Walk: A Renormalisation Group Analysis

We prove that the susceptibility of the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$, in the critical dimension $d=4$, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent 1/4 for the logarithm. The susceptibility has been well understood previously for dimensions $d \geq 5$ using the lace expansion, but the lace expansion does not apply when $d=4$. The proof begins by rewriting the walk two-point function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.

A Renormalisation Group Method. IV. Stability Analysis

This paper is the fourth in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. The third paper in the series presents a perturbative analysis of a supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$. We now present an analysis of the relevant interaction functional of the supersymmetric field theory, which permits a nonperturbative analysis to be carried out in the critical dimension $d = 4$. The results in this paper include: proof of stability of the interaction, estimates which enable control of Gaussian expectations involving both boson and fermion fields, estimates which bound the errors in the perturbative analysis, and a crucial contraction estimate to handle irrelevant directions in the flow of the renormalisation group. These results are essential for the analysis of the general renormalisation group step in the fifth paper in the series.

A general non-Abelian density matrix renormalization group algorithm with application to the C2 dimer.

We extend our previous work [S. Sharma and G. K.-L. Chan, J. Chem. Phys. 136, 124121 (2012)], which described a spin-adapted (SU(2) symmetry) density matrix renormalization group algorithm, to additionally utilize general non-Abelian point group symmetries. A key strength of the present formulation is that the requisite tensor operators are not hard-coded for each symmetry group, but are instead generated on the fly using the appropriate Clebsch-Gordan coefficients. This allows our single implementation to easily enable (or disable) any non-Abelian point group symmetry (including SU(2) spin symmetry). We use our implementation to compute the ground state potential energy curve of the C2 dimer in the cc-pVQZ basis set (with a frozen-core), corresponding to a Hilbert space dimension of 10(12) many-body states. While our calculated energy lies within the 0.3 mEh error bound of previous initiator full configuration interaction quantum Monte Carlo and correlation energy extrapolation by intrinsic scaling calculations, our estimated residual error is only 0.01 mEh, much more accurate than these previous estimates. Due to the additional efficiency afforded by the algorithm, the excitation energies (Te) of eight lowest lying excited states: a(3)Πu, b(3)Σg (-), A(1)Πu, c(3)Σu (+), B(1)Δg, B(') (1)Σg (+), d(3)Πg, and C(1)Πg are calculated, which agree with experimentally derived values to better than 0.06 eV. In addition, we also compute the potential energy curves of twelve states: the three lowest levels for each of the irreducible representations (1)Σg (+), (1)Σu (+), (1)Σg (-), and (1)Σu (-), to an estimated accuracy of 0.1 mEh of the exact result in this basis.

Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model

We study what we call the all-loop anisotropic bosonized Thirring σ-model. This interpolates between the WZW model and the non-Abelian T-dual of the principal chiral model for a simple group. It has an invariance involving the inversion of the matrix parameterizing the coupling constants. We compute the general renormalization group flow equations which assume a remarkably simple form and derive its properties. For symmetric couplings, they consistently truncate to previous results in the literature. One of the examples we provide gives rise to a first order system of differential equations interpolating between the Lagrange and the Darboux–Halphen integrable systems.

Application of Generalized RNG Turbulence Model to Flow in Motored Single-Cylinder PFI Engine

This paper describes a generalized renormalization group (RNG) turbulence model applied to simulate non-reacting flows in an optical single-cylinder PFI engine. A structured computational mesh of the combustion system with complex geometry was generated by ICEM-CFD in conjunction with KIVA-3V code. Turbulent flow in the 4-valve engine, including the exhaust, intake, compression and expansion strokes, was simulated with the standard k – ε and a generalized RNG turbulence model using the KIVA-3V code. Crank angle-resolved results from available experimental data were used as the boundary and initial conditions for the calculation setup. Pressure traces of the simulation results were compared to the phase-averaged measured pressure trace. Predicted radial and vertical velocities along a horizontal line at BDC and radial velocities along the cylinder axis at four crank angles were compared with the experimental measurements. In addition, the velocity field calculated by the generalized RNG turbulence model was compared with experimental data from Particle Image Velocimetry (PIV) measurements. Good agreement was found between the experiment results and simulation results with the generalized RNG turbulence model.

A generalized renormalization group turbulence model and its application to a light-duty diesel engine operating in a low-temperature combustion regime

A generalized renormalization group closure model based on the “dimensionality” of the flow strain rate is proposed and applied to engine flows in this study. In the model, the model coefficients C1, C2, and C3 are constructed as functions of the flow strain rate. Computations were made for compressing/expanding flows in a diesel engine under motoring conditions. It was found that the generalized renormalization group model performs better than the standard renormalization group k−ε model in terms of its predictions of turbulent kinetic energy and model length scales. Computations in a diesel engine operating under low-temperature combustion conditions were also investigated to further assess the performance of the present model. Predictions of the spatial distributions of unburned hydrocarbons were significantly improved and agreed well with available experimental images. Engine-out unburned hydrocarbon emission was predicted to be smaller with the generalized renormalization group model than with the standard renormalization group k−ε model.