Principle Of Minimal Sensitivity Research Articles

Regulator dependence in the functional renormalization group: A quantitative explanation

The search for controlled approximations to study strongly coupled systems remains a very general open problem. Wilson's renormalization group has shown to be an ideal framework to implement approximations going beyond perturbation theory. In particular, the most employed approximation scheme in this context, the derivative expansion, was recently shown to converge and yield accurate and very precise results. However, this convergence strongly depends on the shape of the employed regulator. In this paper we clarify the reason for this dependence and justify, simultaneously, the most commonly employed procedure to fix this dependence, the principle of minimal sensitivity.

Conformal invariance in the nonperturbative renormalization group: A rationale for choosing the regulator.

Field-theoretical calculations performed in an approximation scheme often present a spurious dependence of physical quantities on some unphysical parameters associated with the details of the calculation setup (such as the renormalization scheme or, in perturbation theory, the resummation procedure). In the present article, we propose to reduce this dependence by invoking conformal invariance. Using as a benchmark the three-dimensional Ising model, we show that, within the derivative expansion at order 4, performed in the nonperturbative renormalization group formalism, the identity associated with this symmetry is not exactly satisfied. The calculations which best satisfy this identity are shown to yield critical exponents which coincide to a high accuracy with those obtained by the conformal bootstrap. Additionally, this work gives a strong justification to the success of a widely used criterion for fixing the appropriate renormalization scheme, namely the principle of minimal sensitivity.

Frequency regulators for the nonperturbative renormalization group: A general study and the model A as a benchmark.

We derive the necessary conditions for implementing a regulator that depends on both momentum and frequency in the nonperturbative renormalization-group flow equations of out-of-equilibrium statistical systems. We consider model A as a benchmark and compute its dynamical critical exponent z. This allows us to show that frequency regulators compatible with causality and the fluctuation-dissipation theorem can be devised. We show that when the principle of minimal sensitivity (PMS) is employed to optimize the critical exponents η,ν, and z, the use of frequency regulators becomes necessary to make the PMS a self-consistent criterion.

Asymptotic Functional Form Preservation Method for the Perturbation Theory: Application to Anharmonic Oscillators

The asymptotic functional form preservation method is developed for the perturbation theory to obtain the energy eigenvalues of anharmonic oscillators. The conventional energy perturbative series expansion for the anharmonic oscillator is strongly divergent even if the anharmonicity is small. Employing a transformation containing an unphysical parameter, we analytically continue this series expansion into a new series expansion applicable to all the range of the perturbation parameter. The unphysical parameter is determined by the principle of minimal sensitivity. This new series expansion is reduced to the conventional energy perturbative series expansion for small anharmonicity, and it preserves the correct asymptotic functional form when the perturbation parameter tends to infinity. Then, we use the full‐range energy series expansion to calculate the energy eigenvalues of the anharmonic oscillator. In addition to excellent energy eigenvalues obtained for the oscillator with small and strong anharmonicity, accurate energy eigenvalues can be obtained using the full‐range energy series expansion when the perturbation parameter tends to infinity.

Exploring arbitrarily high orders of optimized perturbation theory in QCD with [formula omitted

Perturbative QCD with nf flavours of massless quarks becomes simple in the hypothetical limit nf→1612, where the leading β-function coefficient vanishes. The Banks–Zaks (BZ) expansion in a0≡8321(1612−nf) is straightforward to obtain from perturbative results in MS‾ or any renormalization scheme (RS) whose nf dependence is ‘regular’. However, ‘irregular’ RS's are perfectly permissible and should ultimately lead to the same BZ results. We show here that the ‘optimal’ RS determined by the Principle of Minimal Sensitivity does yield the same BZ-expansion results when all orders of perturbation theory are taken into account. The BZ limit provides an arena for exploring optimized perturbation theory at arbitrarily high orders. These explorations are facilitated by a ‘master equation’ expressing the optimization conditions in the fixed-point limit. We find an intriguing strong/weak coupling duality a→a⁎2/a about the fixed point a⁎.

Asymptotic Functional Form Preservation Method with Supersymmetric Quantum Mechanics for Anharmonic Oscillators

The asymptotic functional form preservation method is developed in the framework of supersymmetric quantum mechanics to obtain the energies and wave functions of anharmonic oscillators. For each of the ground states in the hierarchy of supersymmetric partner Hamiltonians, we derive a series expansion of the superpotential for . Employing a transformation containing an unphysical parameter, we convert the series expansion of the superpotential into a new series expansion applicable to all the range of x. The unphysical parameter is determined by the principle of minimal sensitivity. Requiring the preservation of the correct asymptotic functional form of the full-range series expansion as x tends to infinity, we obtain the ground and excited state energies. The truncated full-range series expansion for the superpotential provides an approximate analytical expression for the wave function. In addition, several ansatz functional forms are also proposed for the superpotential to obtain the ground and excited state energies with high accuracy. Excellent computational results for the anharmonic oscillator demonstrate that the method proposed here is suitable for solving similar quantum mechanical problems.

Renormalization scheme dependence with renormalization group summation

We consider all radiative corrections to the total electron-positron cross section showing how the renormalization group equation can be used to sum the logarithmic contributions in two ways. First of all, one can sum leading-log etc. contributions. A second summation shows how all logarithmic corrections can be expressed in terms of log-independent contributions. Next, using Stevenson's characterization of renormalization scheme, we examine scheme dependence when using the second way of summing logarithms. The renormalization scheme invariants that arise are then related to those of Stevenson. We consider two choices of renormalization scheme, one resulting in two powers of a running coupling, the second in an infinite series in the two loop running constant. We then establish how the coupling constant arising in one renormalization scheme can be expressed as a power series of the coupling in any other scheme. Next we establish how by using different mass scale at each order of perturbation theory, all renormalization scheme dependence can be absorbed into these mass scales when one uses the second way of summing logarithmic corrections. We then employ this approach to renormalization scheme dependency to the effective potential in a scalar model, showing the result that it is independent of the background field is scheme independent. The way in which the "principle of minimal sensitivity" can be applied after summation is then discussed.

Computing energy eigenvalues of anharmonic oscillators using the double exponential Sinc collocation method

A quantum anharmonic oscillator is defined by the Hamiltonian H=−d2dx2+V(x), where the potential is given by V(x)=∑i=1mcix2i with cm>0. Using the Sinc collocation method combined with the double exponential transformation, we develop a method to efficiently compute highly accurate approximations of energy eigenvalues for anharmonic oscillators. Convergence properties of the proposed method are presented. Using the principle of minimal sensitivity, we introduce an alternate expression for the mesh size for the Sinc collocation method which improves considerably the accuracy in computing eigenvalues for potentials with multiple wells.We apply our method to a number of potentials including potentials with multiple wells. The numerical results section clearly illustrates the high efficiency and accuracy of the proposed method. All our codes are written in Julia and are available upon request.

The numerically optimized regulator and the functional renormalization group

We aim to optimize the functional form of the compactly supported smooth (CSS) regulator within the functional renormalization group (RG), in the framework of bosonized two-dimensional quantum electrodynamics (QED2) and of the three-dimensional O(N = 1) scalar field theory in the local potential approximation (LPA). The principle of minimal sensitivity (PMS) is used for the optimization of the CSS regulator, recovering all the major types of regulators in appropriate limits. Within the investigated class of functional forms, a thorough investigation of the CSS regulator, optimized with two different normalizations within the PMS method, confirms that the functional form of a regulator first proposed by Litim is optimal within the LPA. However, Litim’s exact form leads to a kink in the regulator function. A form of the CSS regulator, numerically close to Litim’s limit while maintaining infinite differentiability, remains compatible with the gradient expansion to all orders. A smooth analytic behavior of the regulator is ensured by a small, but finite value of the exponential fall-off parameter in the CSS regulator. Consequently, a compactly supported regulator, in a parameter regime close to Litim’s optimized form, but regularized with an exponential factor, appears to have favorable properties and could be used to address the scheme dependence of the functional RG, at least within the approximations employed in the studies reported here.

Fixed and unfixed points: Infrared limits in optimized QCD perturbation theory

Perturbative QCD, when optimized by the principle of minimal sensitivity at fourth order, yields finite results for Re+e−(Q) down to Q=0. For two massless flavors (nf=2) this occurs because the couplant “freezes” at a fixed-point of the optimized β function. However, for larger nfʼs, between 6.7 and 15.2, the infrared limit arises by a novel mechanism in which the evolution of the optimized β function with energy Q is crucial. The evolving β function develops a minimum that, as Q→0, just touches the axis at ap (the “pinch point”), while the infrared limit of the optimized couplant is at a larger value, a⋆ (the “unfixed point”). This phenomenon results in R approaching its infrared limit not as a power law, but as R→R⋆−const/|lnQ|2. Implications for the phase structure of QCD as a function of nf are briefly considered.

STUDY OF AMBIGUITIES OF QCD PREDICTIONS AT JET PRODUCTION IN PROTON–PROTON COLLISIONS AT NEXT-TO-LEADING ORDER

The dependence of inclusive one jet differential cross-section on the renormalization and factorization scale is studied. The choice when both scales μr, μfare set equal to jet transversal momentum is taken as a referential point and within a region around this choice, the inclusive one jet differential cross-section is calculated at next-to-leading order. Following the principle of minimal sensitivity (PMS), the region of local stability with respect to both scales is searched for. It is shown that it is well approximated by particular choice [Formula: see text] for total energies 7000 and 14,000 GeV.

Self-consistency requirements of the renormalization group for setting the renormalization scale

In conventional treatments, predictions from fixed-order perturbative QCD calculations cannot be fixed with certainty due to ambiguities in the choice of the renormalization scale as well as the renormalization scheme. In this paper we present a general discussion of the constraints of the renormalization group (RG) invariance on the choice of the renormalization scale. We adopt the RG based equations, which incorporate the scheme parameters, for a general exposition of RG invariance, since they simultaneously express the invariance of physical observables under both the variation of the renormalization scale and the renormalization scheme parameters. We then discuss the self-consistency requirements of the RG, such as reflexivity, symmetry, and transitivity, which must be satisfied by the scale-setting method. The Principle of Minimal Sensitivity (PMS) requires the slope of the approximant of an observable to vanish at the renormalization point. This criterion provides a scheme-independent estimation, but it violates the symmetry and transitivity properties of the RG and does not reproduce the Gell-Mann-Low scale for QED observables. The Principle of Maximum Conformality (PMC) satisfies all of the deductions of the RG invariance - reflectivity, symmetry, and transitivity. Using the PMC, all non-conformal $\{\beta^{\cal R}_i\}$-terms (${\cal R}$ stands for an arbitrary renormalization scheme) in the perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC scales and the resulting finite-order PMC predictions are both to high accuracy independent of the choice of initial renormalization scale, consistent with RG invariance. [...More in the text...]

Nonsensitive Homotopy-Padé Approach: Anharmonic Oscillators

In order to investigate a complicated physical system, it is convenient to consider a simple, easy to solve model, which is chosen to reflect as much physics as possible of the original system, as an ideal approximation. Motivated by this fundamental idea, we propose a novel asymptotic method, the nonsensitive homotopy-Padé approach. In this method, homotopy relations are constructed to link the original system with an ideal, solvable model. An artificial homotopy parameter is introduced to the homotopy relations as the normal perturbation parameter to generate the perturbation series, and is used to implement the Padé approximation. Meanwhile, some other auxiliary nonperturbative parameters, which are used to control the convergence of the perturbation series, are inserted to the approximants, and are fixed via the principle of minimal sensitivity. The method is used to study the eigenvalue problem of the quantum anharmonic oscillators. Highly accurate numerical results show its validity. Possible further studies on this method are also briefly discussed.

Applicability of the linearδexpansion for theλϕ4field theory at finite temperature in the symmetric and broken phases

The thermodynamics of a scalar field with a quartic interaction is studied within the linear $\ensuremath{\delta}$ expansion (LDE) method. Using the imaginary-time formalism the free energy is evaluated up to second order in the LDE. The method generates nonperturbative results that are then used to obtain thermodynamic quantities like the pressure. The phase transition pattern of the model is fully studied, from the broken to the symmetry restored phase. The results are compared with those obtained with other nonperturbative methods and also with ordinary perturbation theory. The results coming from the two main optimization procedures used in conjunction with the LDE method, the principle of minimal sensitivity (PMS) and the fastest apparent convergence (FAC) are also compared with each other and studied in which cases they are applicable or not. The optimization procedures are applied directly to the free energy.

A variational perturbation approximation method in Tsallis non-extensive statistical physics

For the generalized statistical mechanics based on the Tsallis entropy, a variational perturbation approximation method with the principle of minimal sensitivity is developed by calculating the generalized free energy up to the third order in variational perturbation expansion. The approximation up to the first order amounts to a variational approach which covers the variational method developed by E.K. Lenzi, L.C. Malacarne, R.S. Mendes [Phys. Rev. Lett. 80 (1998) 218] and the approximations up to higher orders can systematically improve variational results. As an illustrated example, the generalized free energy for a classical harmonic oscillator (considered in the Lenzi's joint work) are calculated up to the third order, and the resultant approximations up to the first, second, and third orders are numerically compared with the exact result.

Optimization of field-dependent nonperturbative renormalization group flows

We investigate the influence of the momentum cutoff function on the field-dependent nonperturbative renormalization group flows for the three-dimensional Ising model, up to the second order of the derivative expansion. We show that, even when dealing with the full functional dependence of the renormalization functions, the accuracy of the critical exponents can be simply optimized, through the principle of minimal sensitivity, which yields $\ensuremath{\nu}=0.628$ and $\ensuremath{\eta}=0.044$.

Infrared features of the Landau gauge QCD

The infrared features of Landau gauge QCD are studied by the lattice simulation of $\beta=6.0, 16^4, 24^4, 32^4$ and $\beta=6.4, 32^4, 48^4$. We adopt two definitions of the gauge field; 1) $U-$linear 2) $\log U$ and measured the gluon propagator and ghost propagator. Infrared singularity of the gluon propagator is less than that of tree level result but the gluon propagator at 0 momentum remains finite. The infrared singularity of ghost propagator is stronger than the tree level. The QCD running coupling measured by using the gluon propagator and the ghost propagator has a maximum $\alpha_s(p)\simeq 1$ at around $p=0.5GeV$ and decreases as $p$ approaches 0. The data are analyzed in use of formula of the principle of minimal sensitivity(PMS), the effective charge method and the contour-improved perturbation method, which suggest necessity of the resummation of perturbation series in the infrared region together with existence of the infrared fixed point. Kugo-Ojima parameter saturates at about -0.8 in contrast to the theoretically expected value -1.

New optimization methods for converging perturbative series with a field cutoff

We take advantage of the fact that, in $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ problems, a large field cutoff ${\ensuremath{\varphi}}_{\mathrm{max}}$ makes a perturbative series converge toward values exponentially close to the exact values to make optimal choices of ${\ensuremath{\varphi}}_{\mathrm{max}}.$ For a perturbative series terminated at even order, it is in principle possible to adjust ${\ensuremath{\varphi}}_{\mathrm{max}}$ in order to obtain the exact result. For a perturbative series terminated at odd order, the error can only be minimized. It is, however, possible to introduce a mass shift ${m}^{2}\ensuremath{\rightarrow}{m}^{2}(1+\ensuremath{\eta})$ in order to obtain the exact result. We discuss weak and strong coupling methods to determine ${\ensuremath{\varphi}}_{\mathrm{max}}$ and $\ensuremath{\eta}.$ The numerical calculations in this article are performed with a simple integral with one variable. We give arguments indicating that the qualitative features observed should extend to quantum mechanics and quantum field theory. We find that optimization at even order is more efficient than optimization at odd order. We compare our methods with the linear $\ensuremath{\delta}$ expansion (LDE) (combined with the principle of minimal sensitivity), which provides an upper envelope for the accuracy curves of various Pad\'e and Pad\'e-Borel approximants. Our optimization method performs better than the LDE at strong and intermediate coupling, but not at weak coupling, where it appears less robust and subject to further improvements. We also show that it is possible to fix the arbitrary parameter appearing in the LDE using the strong coupling expansion, in order to get accuracies comparable to ours.

Renormalization and factorization scale analysis of production in antiproton-proton collisions

There a sizable and systematic discrepancy between experimental data on the $\bar{b}b$ production in $\bar{\mathrm p}$p, $\gamma$p and $\gamma\gamma$ collisions and existing theoretical calculations within perturbative QCD. Before interpreting this discrepancy as a signal of new physics, it is important to understand quantitatively the ambiguities of conventional calculations. In this paper the uncertainty coming from renormalization and factorization scale dependence of finite order perturbation calculations of the total cross section of $\bar{b}b$ production in $\bar{\mathrm{p}}$p collisions is discussed in detail. It is shown that the mentioned discrepancy is reduced significantly if these scales are fixed via the Principle of Minimal Sensitivity.

Optimization of the derivative expansion in the nonperturbative renormalization group

We study the optimization of nonperturbative renormalization group equations truncated both in fields and derivatives. On the example of the Ising model in three dimensions, we show that the Principle of Minimal Sensitivity can be unambiguously implemented at order $\partial^2$ of the derivative expansion. This approach allows us to select optimized cut-off functions and to improve the accuracy of the critical exponents $\nu$ and $\eta$. The convergence of the field expansion is also analyzed. We show in particular that its optimization does not coincide with optimization of the accuracy of the critical exponents.