Choice Of Renormalization Scale Research Articles

Z-boson hadronic decay width up to {{mathcal {O}}}(alpha _s^4)-order QCD corrections using the single-scale approach of the principle of maximum conformality

In the paper, we study the properties of the Z-boson hadronic decay width by using the mathcal {O}(alpha _s^4)-order quantum chromodynamics (QCD) corrections with the help of the principle of maximum conformality (PMC). By using the PMC single-scale approach, we obtain an accurate renormalization scale-and-scheme independent perturbative QCD (pQCD) correction for the Z-boson hadronic decay width, which is independent to any choice of renormalization scale. After applying the PMC, a more convergent pQCD series has been obtained; and the contributions from the unknown mathcal {O}(alpha _s^5)-order terms are highly suppressed, e.g. conservatively, we have Delta Gamma _{mathrm{Z}}^{mathrm{had}}|^{{{mathcal {O}}}(alpha _s^5)}_{mathrm{PMC}}simeq pm 0.004 MeV. In combination with the known electro-weak (EW) corrections, QED corrections, EW–QCD mixed corrections, and QED–QCD mixed corrections, our final prediction of the hadronic Z decay width is Gamma _{mathrm{Z}}^{mathrm{had}}=1744.439^{+1.390}_{-1.433} MeV, which agrees with the PDG global fit of experimental measurements, 1744.4pm 2.0 MeV.

Revisiting the bottom quark forward\u2013backward asymmetry A_mathrm{{FB}} in electron\u2013positron collisions

The bottom quark forward–backward asymmetry A_mathrm{{FB}} is a key observable in electron–positron collisions at the Z^{0} peak. In this paper, we employ the Principle of Maximum Conformality (PMC) to fix the alpha _s-running behavior of the next-to-next-to-leading order QCD corrections to A_mathrm{{FB}}. The resulting PMC scale for this A_mathrm{{FB}} is an order of magnitude smaller than the conventional choice mu _r=M_Z. This scale has the physically reasonable behavior and reflects the virtuality of its QCD dynamics, which is independent to the choice of renormalization scale. Our analyses show that the effective momentum flow for the bottom quark forward–backward asymmetry should be mu _rll M_Z other than the conventionally suggested mu _r=M_Z. Moreover, the convergence of perturbative QCD series for A_mathrm{{FB}} is greatly improved using the PMC. Our prediction for the bare bottom quark forward–backward asymmetry is refined to be A^{0,b}_mathrm{FB}=0.1004pm 0.0016, which diminishes the well known tension between the experimental determination for this (pseudo) observable and the respective Standard Model fit to 2.1sigma .

Principle of maximum conformality and complete renormalization group improvement approaches to optimize QCD observables: Similarity, differences, and preference

The principle of maximum conformality (PMC) in perturbative QCD predicts energy scales which are independent of the renormalization scheme and choice of the renormalization scale. In this approach the observable can be divided to the conformal and nonconformal sectors. Nonconformal dependence terms are absorbed into the running coupling constant. By absorbing these terms at any desired order, a unique scale can be obtained at that order, and the final result for the observable is independent of the initial choice of renormalization scale and also the employed scheme. This feature is in accordance with the invariant property of renormalization group. On the other side, we consider another approach which is called the complete renormalization group improvement (CORGI) to optimize the perturbative series of QCD observables. In this approach, using the self-consistency principle, all perturbative terms can be reconstructed such that they are expressed in terms of the quantities which do not depend on the renormalization scale. Each reconstructed term in this approach involves a resummation of infinite terms at a specified order. Therefore the conventional perturbative series of QCD observables appear in terms of scheme-invariant quantities. This approach is then expected to get more reliable results with respect to what we obtain in conventional perturbation theory. Considering these two approaches, we are looking to find the differences and similarities and the probable relation which might exist between them. Since the PMC approach is founded on more strong theoretical bases, the comparison between these two approaches will reveal the priority which naturally should be assigned to the PMC approach. We examine some QCD observables like the R-ratio for ${e}^{+}{e}^{\ensuremath{-}}$ annihilation, ${R}_{\ensuremath{\tau}}$ for $\ensuremath{\tau}$ decay, and Higgs decay width to $b\overline{b}$ and gluon-gluon, and we compare the results for these observable in two approaches. For QCD observables in which there are experimental data, the results we obtain from both CORGI and PMC approaches are in good agrement with the experimental data. In other cases, the results from theses two approaches are near to each other; however, they are a little different from the results of the conventional perturbation theory.

Construction of a renormalization group improved effective potential in a two real scalar system

We study the improvement of an effective potential by a renormalization group (RG) equation in a two real scalar system. We clarify the logarithmic structure of the effective potential in this model. Based on the analysis of the logarithmic structure of it, we find that the RG improved effective potential up to |$L$|th-to-leading log order can be calculated by the |$L$|-loop effective potential and |$(L+1)$|-loop |$\beta$| and |$\gamma$| functions. To obtain the RG improved effective potential, we choose the mass eigenvalue as a renormalization scale. If another logarithm at the renormalization scale is large, we decouple the heavy particle from the RG equation and we must modify the RG improved effective potential. In this paper we treat such a situation and evaluate the RG improved effective potential. Although this method was previously developed in a single scalar case, we implement the method in a two real scalar system. The feature of this method is that the choice of renormalization scale does not change even in a calculation of higher leading log order. Following our method one can derive the RG improved effective potential in a multiple scalar model.

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...]

RτAND HIGGS DECAY WIDTH IN CORGI APPROACH, BASED ON THE LEADING-b APPROXIMATION

We consider Adler D-function for the vector and scalar correlators with recent QCD analysis at N4LO approximation. The portion of perturbative coefficients of these observable containing the leading power of b, the first beta-function coefficient, is resummed to all-orders. The factorial behavior of coefficients allows us to use the Borel transformation to reveal the existent singularities as renormalons. CORGI approach is employed to avoid the renormalization dependence of the observable. In addition to the CORGI approach, the calculations of the standard perturbative QCD approach which uses the [Formula: see text] scheme with a physical choice of renormalization scale, are also presented. A comparison between the results of the standard and CORGI approaches for Rτand Higgs decay width is done. The comparison anticipates a better result for the CORGI approach. As an adjunct to these studies, the difference between the all order and fixed order result at N4LO approximation is used to estimate the uncertainty in [Formula: see text] extracted from Rτmeasurements. We find at N4LO approximation a smaller uncertainty with respect to the previous result which has been done at N4LO approximation.

Renormalization group improvement and constituent quark model

The scale dependence in perturbative QCD remains on obstacle to making precise tests of the theory. To over come the scale ambiguity while people are usually using the standard M S ¯ approach with a physical choice of renormalization scale, we try to employ the approach of complete RG-improvement. In this approach all ultraviolet logarithms involving the dimensionful parameter, Q , on which the observable depends are resummed, thereby building the correct Q -dependence. Based on these two approaches, sea quark densities in the nucleon are analyzed. To achieve the asymmetry of these densities, chiral quark model is used. To avoid from the unaccepted Q 2 behavior of sea densities inside the constituent quark, we assume that the free parameter which exists in the vertex function of boson-quark splitting function, is Q 2 -dependence. Using un-symmetrized sea densities of the nucleon, the Gottfried sum rule is calculated in these two approaches. The result of the latter one is very close to the reported experimental value.

MESON CLOUDS AND DRESSED CONSTITUENT QUARKS IN THE COMPLETE RG-IMPROVEMENT APPROACH

Sea quark densities in the nucleon, based on the constituent quark model are analyzed. To model the asymmetry of these densities, the meson cloud or alternatively chiral quark model (χQM) is used. Valence quark densities of the meson which are required to extract the sea quark densities in the constituent quarks are obtained using the phenomenological valon model. In addition to the standard perturbative QCD approach which uses the [Formula: see text] scheme with a physical choice of renormalization scale, the calculations are also performed using the complete RG-improvement (CORGI) approach. To avoid a physically unacceptable Q2 behavior of the sea densities inside the constituent quarks, we assume that the free parameter which exists in the vertex function of the boson–quark splitting function, is Q2-dependent. Using the unsymmetrized sea densities of the nucleon which result from convoluting the constituent density in a nucleon with the quark density in the constituent quark, the Gottfried sum rule (GSR) is calculated using the standard perturbative and CORGI approaches. The CORGI result is closer to the reported experimental value for the GSR. The extracted sea and valence quark density in a nucleon, using χQM and also the CORGI approach, have been compared with available experimental data and what was obtained, based on χQM in the standard approach. This comparison confirms the anticipated better agreement of the CORGI approach with the data.

Quantum fluctuations on a thick de Sitter brane

We investigate quantum fluctuations on a de Sitter (dS) brane, which has its own thickness, in order to examine whether or not the finite thickness of the brane can act as a natural cut-off for the Kaluza–Klein (KK) spectrum. We calculate the amplitude of the KK modes and the bound state by using the zeta function method after a dimensional reduction. We show that the KK amplitude is finite for a given brane thickness and in the thin wall limit the standard surface divergent behavior is recovered. The strength of the divergence in the thin wall limit depends on the number of dimensions, e.g., logarithmic on a two-dimensional brane and quadratic on a four-dimensional brane. We also find that the amplitude of the bound state mode and KK modes depends on the choice of renormalization scale; and for fixed renormalization scales the bound state mode is insensitive to the brane thickness both for two- and four-dimensional dS branes.

Complete two-loop effective potential approximation to the lightest Higgs scalar boson mass in supersymmetry

I present a method for accurately calculating the pole mass of the lightest Higgs scalar boson in supersymmetric extensions of the Standard Model, using a mass-independent renormalization scheme. The Higgs scalar self-energies are approximated by supplementing the exact one-loop results with the second derivatives of the complete two-loop effective potential in Landau gauge. I discuss the dependence of this approximation on the choice of renormalization scale, and note the existence of particularly poor choices which fortunately can be easily identified and avoided. For typical input parameters, the variation in the calculated Higgs mass over a wide range of renormalization scales is found to be of order a few hundred MeV or less, and is significantly improved over previous approximations.

Study of direct photon production at the CERN LHC

Study of direct photon production in high-energy hadronic collisions provides a clean tool for testing the essential validity of perturbative quantum chromodynamics (PQCD) predictions as well as for constraining the gluon distribution of nucleons. These attractive considerations prompted us to study the characteristics of direct photons at CERN LHC energy $(\sqrt{s}=14\mathrm{TeV}).$ In order to validate our simulation results, we first describe the direct photon data at $\sqrt{s}=1.8\mathrm{TeV}$ in the central pseudorapidity $(\ensuremath{\eta})$ region. We used next-to-leading-order (NLO) QCD calculations and leading-order (LO) PYTHIA estimates with the latest parton distribution function, CTEQ5M1. At $\sqrt{s}=14\mathrm{TeV},$ the LO and NLO QCD predictions for direct photon cross section are presented as a function of transverse momentum of photon ${(p}_{T})$ in the kinematical region $20\mathrm{GeV}l{p}_{T}l400\mathrm{GeV}$ and $|\ensuremath{\eta}|l3.$ The sensitivity of the theoretical predictions to the choice of renormalization scales and gluon distributions is also demonstrated. The pseudorapidity $(\ensuremath{\eta})$ and cone size dependence of the direct photon cross section is also discussed.

An investigation of uncertainties in the QCD NLO predictions of the inclusive jet cross section in $\mathbf{\bar{p}p}$ collisions at $\sqrt{s} = 1.8$ TeV and 630 GeV

Uncertainties in the NLO calculation of the inclusive jet cross section due to the choice of renormalization scale, parton distribution functions and clustering algorithm are explored. These are found to be similar in size to the current experimental uncertainties of the measured inclusive jet cross section at DO and CDF.

Erratum: QCD predictions for annihilation decays ofP-wave quarkonia to next-to-leading order inαs[Phys. Rev. D 54, 6850 (1996)

The decay rates of P-wave heavy quarkonia to light hadrons are presented to leading order in $v^2$ and next-to-leading order in $\alpha_s$. They include contributions from both the color-singlet component and the color-octet component of quarkonia. Applying these results to charmonium and using measured decay rates for the $\chi_{c1}$ and $\chi_{c2}$ by E760, we determine the two nonperturbative decay matrix elements, and then predict the hadronic decay rates of $\chi_{c0}$ and $h_c$, and the electromagnetic decay rates of $\chi_{c0}$ and $\chi_{c2}$. The obtained decay rates of $\chi_{c0}\to LH$ and $\chi_{c0}\to\gamma\gamma$ are in agreement with the Crystal Ball result, and also with the new measurement by BES. However, the results for $\Gamma(\chi_{c0}\to LH)$ are dependent on the choice of renormalization scale.

QCD predictions for annihilation decays of P-wave quarkonia to next-to-leading order in alpha s.

The decay rates of {ital P}-wave heavy quarkonia to light hadrons are presented to leading order in {ital v}{sup 2} and next-to-leading order in {alpha}{sub {ital s}}. They include contributions from both the color-singlet component and the color-octet component of quarkonia. Applying these results to charmonium and using measured decay rates for the {chi}{sub {ital c}1} and {chi}{sub {ital c}2} by E760, we determine the two nonperturbative decay matrix elements, and then predict the hadronic decay rates of {chi}{sub {ital c}0} and {ital h}{sub {ital c}}, and the electromagnetic decay rates of {chi}{sub {ital c}0} and {chi}{sub {ital c}2}. The obtained decay rates of {chi}{sub {ital c}0}{r_arrow}{ital LH} and {chi}{sub {ital c}0}{r_arrow}{gamma}{gamma} are in agreement with the Crystal Ball results, and also with the new measurements by BES. However, the results for {Gamma}({chi}{sub {ital c}0}{r_arrow}{ital LH}) are dependent on the choice of renormalization scale. {copyright} {ital 1996 The American Physical Society.}