Understanding parton evolution in matter from renormalization group analysis
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Non-abelian symmetries in tensor networks: A quantum symmetry space approach
A general framework for non-abelian symmetries is presented for matrix-product and tensor-network states in the presence of well-defined orthonormal local as well as effective basis sets. The two crucial ingredients, the Clebsch-Gordan algebra for multiplet spaces as well as the Wigner-Eckart theorem for operators, are accounted for in a natural, well-organized, and computationally straightforward way. The unifying tensor-representation for quantum symmetry spaces, dubbed QSpace, is particularly suitable to deal with standard renormalization group algorithms such as the numerical renormalization group (NRG), the density matrix renormalization group (DMRG), or also more general tensor networks such as the multi-scale entanglement renormalization ansatz (MERA). In this paper, the focus is on the application of the non-abelian framework within the NRG. A detailed analysis is presented for a fully screened spin- 3/2 three-channel Anderson impurity model in the presence of conservation of total spin, particle-hole symmetry, and SU(3) channel symmetry. The same system is analyzed using several alternative symmetry scenarios based on combinations of U(1){sub charge}, SU(2){sub spin}, SU(2){sub charge}, SU(3){sub channel}, as well as the enveloping symplectic Sp(6) symmetry. These are compared in detail, including their respective dramatic gain in numerical efficiency. In the Appendix, finally, an extensive introduction to non-abelian symmetries is given for practical applications, together with simple self-contained numerical procedures to obtain Clebsch-Gordan coefficients and irreducible operators sets. The resulting QSpace tensors can deal with any set of abelian symmetries together with arbitrary non-abelian symmetries with compact, i.e. finite-dimensional, semi-simple Lie algebras. - Highlights: Black-Right-Pointing-Pointer We introduce a transparent framework for non-abelian symmetries in tensor networks. Black-Right-Pointing-Pointer The framework was successfully applied within the numerical renormalization group. Black-Right-Pointing-Pointer Combinations of SU(N{<=}3) and Sp(6) are implemented and discussed in detail. Black-Right-Pointing-Pointer Non-abelian symmetries allow to speed up calculations by orders of magnitude. Black-Right-Pointing-Pointer The framework is equally applicable to other tensor networks such as DMRG or MERA.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
The Principle of Maximum Conformality Correctly Resolves the Renormalization-Scheme-Dependence Problem
In this paper, we clarify a serious misinterpretation and consequent misuse of the Principle of Maximum Conformality (PMC), which also can serve as a mini-review of PMC. In a recently published article, P. M. Stevenson has claimed that “the PMC is ineffective and does nothing to resolve the renormalization-scheme-dependence problem”, concluding incorrectly that the success of PMC predictions is due to the PMC being a “laborious, ad hoc, and back-door” version of the Principle of Minimal Sensitivity (PMS). We show that such conclusions are incorrect, deriving from a misinterpretation of the PMC and an overestimation of the applicability of the PMS. The purpose of the PMC is to achieve precise fixed-order pQCD predictions, free from conventional renormalization schemes and scale ambiguities. We demonstrate that the PMC predictions satisfy all the self-consistency conditions of the renormalization group and standard renormalization-group invariance; the PMC predictions are thus independent of any initial choice of renormalization scheme and scale. The scheme independence of the PMC is also ensured by commensurate scale relations, which relate different observables to each other. Moreover, in the Abelian limit, the PMC dovetails into the well-known Gell-Mann–Low framework, a method universally revered for its precision in QED calculations. Due to the elimination of factorially divergent renormalon terms, the PMC series not only attains a convergence behavior far superior to that of its conventional counterparts but also deftly curtails any residual scale dependence caused by the unknown higher-order terms. This refined convergence, coupled with its robust suppression of residual uncertainties, furnishes a sound and reliable foundation for estimating the contributions from unknown higher-order terms. Anchored in the bedrock of standard renormalization-group invariance, the PMC simultaneously eradicates the factorial divergences and eliminates superfluous systematic errors, which inversely provides a good foundation for achieving high-precision pQCD predictions. Consequently, owing to its rigorous theoretical underpinnings, the PMC is eminently applicable to virtually all high-energy hadronic processes.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Effective and asymptotic criticality of structurally disordered magnets
Changes in magnetic critical behaviour of quenched structurally-disordered magnets are usually exemplified in experiments and in MC simulations by diluted systems consisting of magnetic and non-magnetic components. In our study we aim to show that similar effects can be observed not only for diluted magnets with non-magnetic impurities but may be implemented, e.g., by the presence of two (and more) chemically different magnetic components as well. Therefore we consider a model of the structurally-disordered quenched magnet where all lattice sites are occupied by Ising-like spins of different lengths L. In such a random spin length Ising model, the length L of each spin is a random variable governed by the distribution function p (L). We demonstrate that this model belongs to the universality class of the site-diluted Ising model. This proves that both models are described by the same values of asymptotic critical exponents. However, their effective critical behaviour differs. As a case study, we consider a quenched mixture of two different magnets with values of elementary magnetic moments L 1 = 1 and L 2 = s, and of concentration c and 1 - c, correspondingly. We apply field-theoretical renormalization group approach to analyse the renormalization group flow for different initial conditions, triggered by s and c, and to calculate effective critical exponents further away from the fixed points of the renormalization group transformation. We show how the effective exponents are governed by difference in properties of the magnetic components.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Transverse momentum dependent fragmenting jet functions with applications to quarkonium production
We introduce the transverse momentum dependent fragmenting jet function (TMDFJF), which appears in factorization theorems for cross sections for jets with an identified hadron. These are functions of z, the hadron’s longitudinal momentum fraction, and transverse momentum, p ⊥ , relative to the jet axis. In the framework of Soft-Collinear Effective Theory (SCET) we derive the TMDFJF from both a factorized SCET cross section and the TMD fragmentation function defined in the literature. The TMDFJFs are factorized into distinct collinear and soft-collinear modes by matching onto SCET + . As TMD calculations contain rapidity divergences, both the renormalization group (RG) and rapidity renormalization group (RRG) must be used to provide resummed calculations with next-to-leading-logarithm prime (NLL’) accuracy. We apply our formalism to the production of J/ψ within jets initiated by gluons. In this case the TMDFJF can be calculated in terms of NRQCD (Non-relativistic quantum chromodynamics) fragmentation functions. We find that when the J/ψ carries a significant fraction of the jet energy, the p T and z distributions differ for different NRQCD production mechanisms. Another observable with discriminating power is the average angle that the J/ψ makes with the jet axis.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
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. Finally, power-law clocks are introduced to examine multi-scaling behavior. Several examples of these ideas are presented and discussed.
58 GEOSCIENCES↗
High precision tests of QCD without scale or scheme ambiguities: The 40th anniversary of the Brodsky–Lepage–Mackenzie method
A key issue in making precise predictions in QCD is the uncertainty in setting the renormalization scale μ r and thus determining the correct values of the QCD running coupling α s (μ r ) at each order in the perturbative expansion of a QCD observable. It has often been conventional to simply set the renormalization scale to the typical scale of the process Q and vary it in the range μ r $\in$ [Q/2, 2Q] in order to estimate the theoretical error. This is the practice of Conventional Scale Setting (CSS). The resulting CSS prediction will however depend on the theorist’s choice of renormalization scheme and the resulting pQCD series will diverge factorially. It will also disagree with renormalization scale setting used in QED and electroweak theory thus precluding grand unification. A solution to the renormalization scale-setting problem is offered by the Principle of Maximum Conformality (PMC), which provides a systematic way to eliminate the renormalization scale-and-scheme dependence in perturbative calculations. The PMC method has rigorous theoretical foundations, it satisfies Renormalization Group Invariance (RGI) and preserves all self-consistency conditions derived from the renormalization group. The PMC cancels the renormalon growth, reduces to the Gell-Mann–Low scheme in the N c → 0 Abelian limit and leads to scale- and scheme-invariant results. The PMC has now been successfully applied to many high-energy processes. In this article we summarize recent developments and results in solving the renormalization scale and scheme ambiguities in perturbative QCD. In particular, we present a recently developed method the PMC ∞ and its applications, comparing the results with CSS. The method preserves the property of renormalizable SU(N)/U(1) gauge theories defined as Intrinsic Conformality (iCF). This property underlies the scale invariance of physical observables and leads to a remarkably efficient method to solve the conventional renormalization scale ambiguity at every order in pQCD. This new method reflects the underlying conformal properties displayed by pQCD at NNLO, eliminates the scheme dependence of pQCD predictions and is consistent with the general properties of the PMC. A new method to identify conformal and β-terms, which can be applied either to numerical or to theoretical calculations is also shown. We present results for the thrust and C-parameter distributions in e + e - annihilation showing errors and comparison with the CSS. We also show results for a recent innovative comparison between the CSS and the PMC ∞ applied to the thrust distribution investigating both the QCD conformal window and the QED N c → 0 limit. In order to determine the thrust distribution along the entire renormalization group flow from the highest energies to zero energy, we consider the number of flavors near the upper boundary of the conformal window. In this flavor-number regime the theory develops a perturbative infrared interacting fixed point. These results show that PMC ∞ leads to higher precision and introduces new interesting features in the PMC. In fact, this method preserves with continuity the position of the peak, showing perfect agreement with the experimental data already at NNLO. We also show a detailed comparison of the PMC ∞ with the other PMC approaches: the multi-scale-setting approach (PMCm) and the single-scale-setting approach (PMCs) by comparing their predictions for three important fully integrated quantities R e+e- , R$_Τ$ and Γ (H→$b\bar{b}$) up to the four-loop accuracy.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
The long flow to freedom
Two-dimensional field theories do not have a moduli space of vacua. Instead, it is common that their low-energy behavior is a sigma model with a target space. When this target space is compact its renormalization group flow is standard. When it is non-compact the continuous spectrum of operators can change the qualitative behavior. Here we discuss two-dimensional gauge theories with N = (2,2) supersymmetry. We focus on two specific theories, for which we argue that they flow to free chiral multiplets at low energies: the U(1) gauge theory with one flavor (two chiral superfields with charges plus and minus one) and a non-zero Fayet-Iliopoulos term, and pure SU( N ) gauge theories. We argue that the renormalization group flow of these theories has an interesting order of limits issue. Holding the position on the target space fixed, the space flattens out under the renormalization group. On the other hand, if we first go to infinity on the target space and then perform the renormalization group, we always have a non-trivial space, e.g. a cone with a deficit angle. We explain how to interpret low-energy dualities between theories with non-compact target spaces. As a result, we expect a similar qualitative behavior also for other non-compact sigma models, even when they do not flow to free theories.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Spectroscopic accuracy directly from quantum chemistry: Application to ground and excited states of beryllium dimer
We combine explicit correlation via the canonical transcorrelation approach with the density matrix renormalization group and initiator full configuration interaction quantum Monte Carlo methods to compute a near-exact beryllium dimer curve, without the use of composite methods. In particular, our direct density matrix renormalization group calculations produce a well-depth of D{sub e} = 931.2 cm{sup −1} which agrees very well with recent experimentally derived estimates D{sub e} = 929.7±2 cm{sup −1} [J. M. Merritt, V. E. Bondybey, and M. C. Heaven, Science 324, 1548 (2009)] and D{sub e}= 934.6 cm{sup −1} [K. Patkowski, V. Špirko, and K. Szalewicz, Science 326, 1382 (2009)], as well the best composite theoretical estimates, D{sub e} = 938±15 cm{sup −1} [K. Patkowski, R. Podeszwa, and K. Szalewicz, J. Phys. Chem. A 111, 12822 (2007)] and D{sub e}=935.1±10 cm{sup −1} [J. Koput, Phys. Chem. Chem. Phys. 13, 20311 (2011)]. Our results suggest possible inaccuracies in the functional form of the potential used at shorter bond lengths to fit the experimental data [J. M. Merritt, V. E. Bondybey, and M. C. Heaven, Science 324, 1548 (2009)]. With the density matrix renormalization group we also compute near-exact vertical excitation energies at the equilibrium geometry. These provide non-trivial benchmarks for quantum chemical methods for excited states, and illustrate the surprisingly large error that remains for 1 {sup 1}Σ{sub g}{sup −} state with approximate multi-reference configuration interaction and equation-of-motion coupled cluster methods. Overall, we demonstrate that explicitly correlated density matrix renormalization group and initiator full configuration interaction quantum Monte Carlo methods allow us to fully converge to the basis set and correlation limit of the non-relativistic Schrödinger equation in small molecules.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Importance of proper renormalization scale-setting for QCD testing at colliders
A primary problem affecting perturbative quantum chromodynamic (pQCD) analyses is the lack of a method for setting the QCD running-coupling renormalization scale such that maximally precise fixed-order predictions for physical observables are obtained. The Principle of Maximum Conformality (PMC) eliminates the ambiguities associated with the conventional renormalization scale-setting procedure, yielding predictions that are independent of the choice of renormalization scheme. The QCD coupling scales and the effective number of quark flavors are set order-by-order in the pQCD series. The PMC has a solid theoretical foundation, satisfying the standard renormalization group invariance condition and all of the self-consistency conditions derived from the renormalization group. The PMC scales at each order are obtained by shifting the arguments of the strong force coupling constant αs to eliminate all non-conformal {βi} terms in the pQCD series. The {βi} terms are determined from renormalization group equations without ambiguity. The correct behavior of the running coupling at each order and at each phase-space point can then be obtained. The PMC reduces in the N C → 0 Abelian limit to the Gell-Mann-Low method. In this brief report, we summarize the results of our recent application of the PMC to a number of collider processes, emphasizing the generality and applicability of this approach. A discussion of hadronic Z decays shows that, by applying the PMC, one can achieve accurate predictions for the total and separate decay widths at each order without scale ambiguities. We also show that, if one employs the PMC to determine the top-quark pair forward-backward asymmetry at the next-to-next-to-leading order level, one obtains a comprehensive, self-consistent pQCD explanation for the Tevatron measurements of the asymmetry. This accounts for the “increasing-decreasing” behavior observed by the D0 collaboration for increasing tt¯ invariant mass. At lower energies, the angular distributions of heavy quarks can be used to obtain a direct determination of the heavy quark potential. A discussion of the angular distributions of massive quarks and leptons is also presented, including the fermionic component of the two-loop corrections to the electromagnetic form factors. Furthermore, these results demonstrate that the application of the PMC systematically eliminates a major theoretical uncertainty for pQCD predictions, thus increasing collider sensitivity to possible new physics beyond the Standard Model.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Extending the predictive power of perturbative QCD
The predictive power of perturbative QCD (pQCD) depends on two important issues: (1) how to eliminate the renormalization scheme-and-scale ambiguities at fixed order, and (2) how to reliably estimate the contributions of unknown higher-order terms using information from the known pQCD series. The Principle of Maximum Conformality (PMC) satisfies all of the principles of the renormalization group and eliminates the scheme-and-scale ambiguities by the recursive use of the renormalization group equation to determine the scale of the QCD running coupling αs at each order. Moreover, the resulting PMC predictions are independent of the choice of the renormalization scheme, satisfying the key principle of renormalization group invariance. In this paper, we show that by using the conformal series derived using the PMC single-scale procedure, in combination with the Padé Approximation Approach (PAA), one can achieve quantitatively useful estimates for the unknown higher-order terms from the known perturbative series. We illustrate this procedure for three hadronic observables R e + e – , R τ , and Γ( H → bb¯) which are each known to 4 loops in pQCD. We show that if the PMC prediction for the conformal series for an observable (of leading order α p s ) has been determined at order αns, then the [ ] = [0/n–p] Padé series provides quantitatively useful predictions for the higher-order terms. We also show that the PMC + PAA predictions agree at all orders with the fundamental, scheme-independent Generalized Crewther relations which connect observables, such as deep inelastic neutrino-nucleon scattering, to hadronic e + e – annihilation. Furthermore, by using the combination of the PMC series and the Padé method, the predictive power of pQCD theory can be greatly improved.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Quantum field theory for the chiral clock transition in one spatial dimension
We describe the quantum phase transition in the N-state chiral clock model in spatial dimension d = 1. With couplings chosen to preserve time-reversal and spatial inversion symmetries, such a model is in the universality class of recent experimental studies of the ordering of pumped Rydberg states in a one-dimensional chain of trapped ultracold alkali atoms. For such couplings and N = 3, the clock model is expected to have a direct phase transition from a gapped phase with a broken global Z N symmetry, to a gapped phase with the Z N symmetry restored. The transition has dynamical critical exponent z ≠ 1, and so cannot be described by a relativistic quantum field theory. We use a lattice duality transformation to map the transition onto that of a Bose gas in d = 1, involving the onset of a single-boson condensate in the background of a higher-dimensional N-boson condensate. Here, we present a renormalization group analysis of the strongly coupled field theory for the Bose gas transition in an expansion in 2 – d, with 4 – N chosen to be of order 2 – d. At two-loop order, we find a regime of parameters with a renormalization group fixed point which can describe a direct phase transition. We also present numerical density-matrix renormalization group studies of lattice chiral clock and Bose gas models for N = 3, finding good evidence for a direct phase transition, and obtain estimates for z and the correlation length exponent ν.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
A collinear perspective on the Regge limit
The high energy (Regge) limit provides a playground for understanding all loop structures of scattering amplitudes, and plays an important role in the description of many phenomenologically relevant cross-sections. While well understood in the planar limit, the structure of non-planar corrections introduces many fascinating complexities, for which a general organizing principle is still lacking. We study the structure of multi-reggeon exchanges in the context of the effective field theory for forward scattering, and derive their factorization into collinear operators (impact factors) and soft operators. We derive the structure of the renormalization group consistency equations in the effective theory, showing how the anomalous dimensions of the soft operators are related to those of the collinear operators, allowing us to derive renormalization group equations in the Regge limit purely from a collinear perspective. The rigidity of the consistency equations provides considerable insight into the all orders organization of Regge amplitudes in the effective theory, as well as its relation to other approaches. Along the way we derive a number of technical results that improve the understanding of the effective theory. We illustrate this collinear perspective by re-deriving all the standard BFKL equations for two-Glauber exchange from purely collinear calculations, and we show that this perspective provides a number of conceptual and computational advantages as compared to the standard view from soft or Glauber physics. We anticipate that this formulation in terms of collinear operators will enable a better understanding of the relation between BFKL and DGLAP in gauge theories, and facilitate the analysis of renormalization group evolution equations describing Reggeization beyond next-to-leading order.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Nonequilibrium spintronic transport through Kondo impurities
In this work we analyze the nonequilibrium transport through a quantum impurity (quantum dot or molecule) attached to ferromagnetic leads by using a hybrid numerical renormalization group–time-dependent density matrix renormalization group thermofield quench approach. For this, we study the bias dependence of the differential conductance through the system, which shows a finite zero-bias peak, characteristic of the Kondo resonance and reminiscent of the equilibrium local density of states. In the nonequilibrium settings, the resonance in the differential conductance is also found to decrease with increasing the lead spin polarization. The latter induces an effective exchange field that lifts the spin degeneracy of the dot level. Therefore, as we demonstrate, the Kondo resonance can be restored by counteracting the exchange field with a finite external magnetic field applied to the system. Lastly, we investigate the influence of temperature on the nonequilibrium conductance, focusing on the split Kondo resonance. Our work thus provides an accurate quantitative description of the spin-resolved transport properties relevant for quantum dots and molecules embedded in magnetic tunnel junctions.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
The ϵ Expansion and Universality in Three Dimensions
It has been observed that the classification into universality classes of critical behavior, as established by perturbative renormalization group in the vicinity of four or six dimensions of space by the epsilon expansion, remains valid down to three dimensions in all known cases, even when perturbative renormalization group fails in lower dimensions. In this paper we argue that this classification into universality classes remains true in lower dimensions of space, even when perturbative renormalization group fails, because of the well known phenomenon of eigenvalue repulsion.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Incorporating Brueckner-Hartree-Fock correlations in energy density functionals
Recently, a microscopically motivated nuclear energy density functional was derived by applying the density-matrix expansion to the Hartree-Fock (HF) energy obtained from long-range chiral effective-field theory two- and three-nucleon interactions. However, the HF approach cannot account for all many-body correlations. One class of correlations is included by Brueckner-Hartree-Fock (BHF) theory, which gives an improved definition of the one-body HF potential by replacing the interaction by a reaction matrix G. In this paper, we find that the difference between the G matrix and the similarity renormalization group evolved nucleon-nucleon potential V SRG can be well accounted for by a truncated series of contact terms. This is consistent with renormalization-group decoupling generating a series of counterterms as short-distance physics is integrated out. Here, the coefficients C n of the power series expansion Σ C n q n for the counterterms are examined for two potentials at different renormalization-group resolutions and at a range of densities. The success of this expansion for G–V SRG means we can apply the density-matrix expansion at the HF level with low-momentum interactions and density-dependent zero-range interactions to model BHF correlations.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Scale-fixed predictions for γ + η c production in electron-positron collisions at NNLO in perturbative QCD
In the paper, we present QCD predictions for γ + η c production at an electron-positron collider up to next-to-next-to-leading order (NNLO) accuracy without renormalization scale ambiguities. The NNLO total cross-section for e + + e – → γ + η c using the conventional scale-setting approach has large renormalization scale ambiguities, usually estimated by choosing the renormalization scale to be the e + e – center-of-mass collision energy $\sqrt{s}$. The Principle of Maximum Conformality (PMC) provides a systematic way to eliminate such renormalization scale ambiguities by summing the nonconformal β contributions into the QCD coupling α s ( Q 2 ). The renormalization group equation then sets the value of α s for the process. The PMC renormalization scale reflects the virtuality of the underlying process, and the resulting predictions satisfy all of the requirements of renormalization group invariance, including renormalization scheme invariance. After applying the PMC, we obtain a renormalization scale-and-scheme independent prediction, σ | NNLO,PMC ≃ 41.18 fb for $\sqrt{s}$=10.6 GeV. The resulting pQCD series matches the series for conformal theory and thus has no divergent renormalon contributions. The large K factor which contributes to this process reinforces the importance of uncalculated NNNLO and higher-order terms. Using the PMC scale-and-scheme independent conformal series and the Padé approximation approach, we predict σ | NNNLO,PMC+Pade ≃ 18.99 fb, which is consistent with the recent BELLE measurement ${\sigma}^{\mathrm{obs}}={16.58}_{-9.93}^{+10.51}$ fb at $\sqrt{s}$ ≃ 10.6 GeV. This procedure also provides a first estimate of the NNNLO contribution.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Nonperturbative renormalization of scalar quantum electrodynamics in d=3
For scalar quantum electrodynamics on a three-dimensional toroidal lattice with a fine lattice spacing, we consider the renormalization problem of choosing counter terms depending on the lattice spacing, so that the theory stays finite as the spacing goes to zero. We employ a renormalization group method which analyzes the flow of the mass and the vacuum energy as a problem in discrete dynamical systems. The main result is that counter terms can be chosen so that at the end of the iteration these quantities take preassigned values. No use is made of perturbation theory. The renormalization group transformations are defined with bounded fields, an approximation which can be justified in Balaban’s approach to the renormalization group.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗