arXiv subjects

Einan Gardi

Publications and source records attributed to Einan Gardi.

98 records · Page 6Linked to original sources

arXiv1999

The conformal window in QCD and supersymmetric QCD

In both QCD and supersymmetric QCD (SQCD) with N_f flavors there are conformal windows where the theory is asymptotically free in the ultraviolet while the infrared physics is governed by a non-trivial fixed-point. In SQCD, the lower N_f boundary of the conformal window, below which the theory is confining is well understood thanks to duality. In QCD there is just a sufficient condition for confinement based on superconvergence. Studying the Banks-Zaks expansion and analyzing the conditions for the perturbative coupling to have a causal analyticity structure, it is shown that the infrared fixed-point in QCD is perturbative in the entire conformal window. This finding suggests that there can be no analog of duality in QCD. On the other hand in SQCD the infrared region is found to be strongly coupled in the lower part of the conformal window, in agreement with duality. Nevertheless, we show that it is possible to interpolate between the Banks-Zaks expansions in the electric and magnetic theories, for quantities that can be calculated perturbatively in both. This interpolation is explicitly demonstrated for the critical exponent that controls the rate at which a generic physical quantity approaches the fixed-point.

hep-th↗

arXiv1998

Relations between Observables and the Infrared Fixed-Point in QCD

We investigate the possibility that alpha_s freezes as function of N_f within perturbation theory. We use two approaches -- direct search for a zero in the effective-charge (ECH) beta function, and the Banks-Zaks (BZ) expansion. We emphasize the fundamental difference between quantities with space-like vs. those with time-like momentum. We show that within the ECH approach several space-like quantities exhibit similar behavior. In general the 3-loop ECH beta functions can lead to freezing for N_f\gsim 5, but higher-order calculations are essential for a conclusive answer. The BZ expansion behaves differently for different observables. Assuming that the existence of a fixed point requires convergence of the BZ expansion for any observable, we can be pretty sure that there is no fixed point for N_f\lsim 12. The consequences of the Crewther relation concerning perturbative freezing are analyzed. We also emphasize that time-like quantities have a consistent infrared limit only when the corresponding space-like effective charge has one. We show that perturbative freezing can lead to an analyticity structure in the complex momentum-squared plane that is consistent with causality.

hep-ph↗

arXiv1998

Can the QCD running coupling have a causal analyticity structure?

Solving the QCD renormalization group equation at the 2-loop and 3-loop orders we obtain explicit expressions for the coupling as a function of the scale in terms of the Lambert W function. We study the nature of the ``Landau singularities'' in the complex Q^2 plane and show that perturbative freezing can lead, in certain cases, to an analyticity structure that is consistent with causality. We analyze the Analytic Perturbation Theory (APT) approach which is intended to remove the ``Landau singularities'', and show that at 2-loops it is uniquely defined in terms of the Lambert W function, and that, depending on the value of the first two beta function coefficients beta_0 and beta_1, it is either consistent with perturbative freezing (for beta_1 < -beta_0^2) with an infrared limit of -beta_0/beta_1 or leads to a non-perturbative infrared coupling with a limit of 1/beta_0 (for beta_1 > -beta_0^2). The possibility of a causal perturbative coupling is in accordance with the idea that a purely perturbative Banks-Zaks phase with an infrared fixed-point exists in QCD if the number of flavours (N_f) is increased. The causality condition implies that the perturbative phase is realized for N_f \geq 10.

hep-ph↗

arXiv1997

Resummation of perturbative QCD by Pade approximants

In this lecture I present some of the new developments concerning the use of Pade Approximants (PA's) for resumming perturbative series in QCD. It is shown that PA's tend to reduce the renormalization scale and scheme dependence as compared to truncated series. In particular it is proven that in the limit where the $β$ function is dominated by the 1-loop contribution, there is an exact symmetry that guarantees invariance of diagonal PA's under changing the renormalization scale. In addition it is shown that in the large $β_0$ approximation diagonal PA's can be interpreted as a systematic method for approximating the flow of momentum in Feynman diagrams. This corresponds to a new multiple scale generalization of the Brodsky-Lepage-Mackenzie (BLM) method to higher orders. I illustrate the method with the Bjorken sum rule and the vacuum polarization function.

hep-ph↗

arXiv1997

Pade Approximants, Optimal Renormalization Scales, and Momentum Flow in Feynman Diagrams

We show that the Pade Approximant (PA) approach for resummation of perturbative series in QCD provides a systematic method for approximating the flow of momentum in Feynman diagrams. In the large-$β_0$ limit, diagonal PA's generalize the Brodsky-Lepage-Mackenzie (BLM) scale-setting method to higher orders in a renormalization scale- and scheme-invariant manner, using multiple scales that represent Neubert's concept of the distribution of momentum flow through a virtual gluon. If the distribution is non-negative, the PA's have only real roots, and approximate the distribution function by a sum of delta-functions, whose locations and weights are identical to the optimal choice provided by the Gaussian quadrature method for numerical integration. We show how the first few coefficients in a perturbative series can set rigorous bounds on the all-order momentum distribution function, if it is positive. We illustrate the method with the vacuum polarization function and the Bjorken sum rule computed in the large-$β_0$ limit.

hep-ph↗

arXiv1996

Why Padé Approximants reduce the Renormalization-Scale dependence in QFT?

We prove that in the limit where the beta function is dominated by the 1-loop contribution (``large beta_0 limit'') diagonal Padé Approximants (PA's) of perturbative series become exactly renormalization scale (RS) independent. This symmetry suggest that diagonal PA's are resumming correctly contributions from higher order diagrams that are responsible for the renormalization of the coupling-constant. Non-diagonal PA's are not exactly invariant, but generally reduce the RS dependence as compared to partial-sums. In physical cases, higher-order corrections in the beta function break the symmetry softly, introducing a small scale and scheme dependence. We also compare the Padé resummation with the BLM method. We find that in the large-N_f limit using the BLM scale is identical to resumming the series by a $x[0/n]$ non-diagonal PA.

hep-ph↗

arXiv1996

Renormalization-Scheme Dependence of Pade Summation in QCD

We study the renormalization-scheme (RS) dependence of Pade Approximants (PA's), and compare them with the Principle of Minimal Sensitivity (PMS) and the Effective Charge (ECH) approaches. Although the formulae provided by the PA, PMS and ECH predictions for higher-order terms in a QCD perturbation expansion differ in general, their predictions can be very close numerically for a wide range of renormalization schemes. Using the Bjorken sum rule as a test case, we find that Pade Summation (PS) reduces drastically the RS dependence of the Bjorken effective charge. We use these results to estimate the theoretical error due to the choice of RS in the extraction of $α_s$ from the Bjorken sum rule, and use the available data at $Q^2=3 GeV^2$ to estimate $α_s(M_Z) = 0.117^{+0.004}_{-0.007} \pm 0.002$, where the first error is experimental, and the second is theoretical.

hep-ph↗

arXiv1995

Pade Approximants, Borel Transforms and Renormalons: the Bjorken Sum Rule as a Case Study

We prove that Pade approximants yield increasingly accurate predictions of higher-order coefficients in QCD perturbation series whose high-order behaviour is governed by a renormalon. We also prove that this convergence is accelerated if the perturbative series is Borel transformed. We apply Pade approximants and Borel transforms to the known perturbative coefficients for the Bjorken sum rule. The Pade approximants reduce considerably the renormalization-scale dependence of the perturbative correction to the Bjorken sum rule. We argue that the known perturbative series is already dominated by an infra-red renormalon, whose residue we extract and compare with QCD sum-rule estimates of higher-twist effects. We use the experimental data on the Bjorken sum rule to extract $α_s(M_Z^2) = 0.116_{-0.006}^{+0.004}$, including theoretical errors due to the finite order of available perturbative QCD calculations, renormalization-scale dependence and higher-twist effects.

hep-ph↗