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Register a new userRenormalization Group Invariance and Optimal QCD Renormalization Scale-Setting
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A valid prediction for a physical observable from quantum field theory should be independent of the choice of renormalization scheme -- this is the primary requirement of renormalization group invariance (RGI). Satisfying scheme invariance is a challenging problem for perturbative QCD (pQCD), since a truncated perturbation series does not automatically satisfy the requirements of the renormalization group. Two distinct approaches for satisfying the RGI principle have been suggested in the literature. One is the Principle of Maximum Conformality (PMC) in which the terms associated with the $beta$-function are absorbed into the scale of the running coupling at each perturbative order; its predictions are scheme and scale independent at every finite order. The other approach is the Principle of Minimum Sensitivity (PMS), which is based on local RGI; the PMS approach determines the optimal renormalization scale by requiring the slope of the approximant of an observable to vanish. In this paper, we present a detailed comparison of the PMC and PMS procedures by analyzing two physical observables $R_{e+e-}$ and $Gamma(Hto bbar{b})$ up to four-loop order in pQCD. At the four-loop level, the PMC and PMS predictions for both observables agree within small errors with those of conventional scale setting assuming a physically-motivated scale, and each prediction shows small scale dependences. However, the convergence of the pQCD series at high orders, behaves quite differently: The PMC displays the best pQCD convergence since it eliminates divergent renormalon terms; in contrast, the convergence of the PMS prediction is questionable, often even worse than the conventional prediction based on an arbitrary guess for the renormalization scale. ......
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A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale $mu$ of the running coupling $alpha_s(mu^2).$ The purpose of the running coupling in any gauge theory is to sum all terms involving the $beta$ function; in fact, when the renormalization scale is set properly, all non-conformal $beta e 0$ terms in a perturbative expansion arising from renormalization are summed into the running coupling. The remaining terms in the perturbative series are then identical to that of a conformal theory; i.e., the corresponding theory with $beta=0$. The resulting scale-fixed predictions using the principle of maximum conformality (PMC) are independent of the choice of renormalization scheme -- a key requirement of renormalization group invariance. The results avoid renormalon resummation and agree with QED scale-setting in the Abelian limit. The PMC is also the theoretical principle underlying the BLM procedure, commensurate scale relations between observables, and the scale-setting method used in lattice gauge theory. The number of active flavors $n_f$ in the QCD $beta$ function is also correctly determined. We discuss several methods for determining the PMC scale for QCD processes. We show that a single global PMC scale, valid at leading order, can be derived from basic properties of the perturbative QCD cross section. The elimination of the renormalization scale ambiguity and the scheme dependence using the PMC will not only increase the precision of QCD tests, but it will also increase the sensitivity of collider experiments to new physics beyond the Standard Model.
We introduce a systematic approach for the resummation of perturbative series which involve large logarithms not only due to large invariant mass ratios but large rapidities as well. Series of this form can appear in a variety of gauge theory observables. The formalism is utilized to calculate the jet broadening event shape in a systematic fashion to next to leading logarithmic order. An operator definition of the factorized cross section as well as a closed form of the next-to leading log cross section are presented. The result agrees with the data to within errors.
We identify a property of renormalizable SU(N)/U(1) gauge theories, the intrinsic Conformality ($iCF$), which 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: the PMC$_infty$. 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 (Principle of Maximum Conformality). We introduce a new method to identify conformal and $beta$-terms which can be applied either to numerical or to theoretical calculations. We illustrate the PMC$_infty$ for the thrust and C-parameter distributions in $e^+ e^-$ annihilation and then we show how to apply this new method to general observables in QCD. We point out how the implementation of the PMC$_infty$ can significantly improve the precision of pQCD predictions; its implementation in multi-loop analysis also simplifies the calculation of higher orders corrections in a general renormalizable gauge theory.
We use a renormalization group method to treat QCD-vacuum behavior specially closer to the regime of asymptotic freedom. QCD-vacuum behaves effectively like a paramagnetic system of a classical theory in the sense that virtual color charges (gluons) emerges in it as a spin effect of a paramagnetic material when a magnetic field aligns their microscopic magnetic dipoles. Due to that strong classical analogy with the paramagnetism of Landaus theory,we will be able to use a certain Landau effective action without temperature and phase transition for just representing QCD-vacuum behavior at higher energies as being magnetization of a paramagnetic material in the presence of a magnetic field $H$. This reasoning will allow us to apply Thompsons approach to such an action in order to extract an effective susceptibility ($chi>0$) of QCD-vacuum. It depends on logarithmic of energy scale $u$ to investigate hadronic matter. Consequently we are able to get an ``effective magnetic permeability ($mu>1$) of such a paramagnetic vacuum. Actually,as QCD-vacuum must obey Lorentz invariance,the attainment of $mu>1$ must simply require that the effective electrical permissivity is $epsilon<1$ in such a way that $muepsilon=1$ ($c^2=1$). This leads to the anti-screening effect where the asymptotic freedom takes place. We will also be able to extend our investigation to include both the diamagnetic fermionic properties of QED-vacuum (screening) and the paramagnetic bosonic properties of QCD-vacuum (anti-screening) into the same formalism by obtaining a $beta$-function at 1 loop,where both the bosonic and fermionic contributions are considered.
We summarize results for local and global properties of the effective potential for the Higgs boson obtained from the functional renormalization group, which allows to describe the effective potential as a function of both scalar field amplitude and RG scale. This sheds light onto the limitations of standard estimates which rely on the identification of the two scales and helps clarifying the origin of a possible property of meta-stability of the Higgs potential. We demonstrate that the inclusion of higher-dimensional operators induced by an underlying theory at a high scale (GUT or Planck scale) can relax the conventional lower bound on the Higgs mass derived from the criterion of absolute stability.
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