QCD Running Coupling Constant Research Articles

Gluonic and leptonic decays of heavy quarkonia and the determination of alpha s(mc) and alpha s(mb).

QCD running coupling constant $\alpha_s(m_c)$ and $\alpha_s(m_b)$ are determined from heavy quarkonia $c\overline{c}$ and $b\overline{b}$ decays. The decay rates of $V\rightarrow 3g$ and $V\rightarrow e^+ e^-$ for $V=J/\psi$ and $\Upsilon$ are estimated by taking into account both relativistic and QCD radiative corrections. The decay amplitudes are derived in the Bethe-Salpeter formalism, and the decay rates are estimated by using the meson wavefunctions which are obtained with a QCD-inspired inter-quark potential. For the $V\rightarrow 3g$ decay we find the relativistic correction to be very large and to severely suppress the decay rate. Using the experimental values of ratio $R_g\equiv \frac {\Gamma (V\longrightarrow 3g)}% {\Gamma (V\longrightarrow e^{+}e^{-})}\approx 10,~32 $ for $V=J/\psi, ~\Upsilon$ respectively, and the calculated widths , we find $\alpha_{s}(m_c)=0.29\pm 0.02 $ and $\alpha_s(m_b)=0.20\pm 0.02$. These values for the QCD running coupling constant are substantially enhanced, as compared with the ones obtained without relativistic corrections, and are consistent with the QCD scale parameter $\Lambda_{\overline {MS}}^{(4)}% \approx 200MeV$. We also find that these results are mainly due to kinematic corrections and not sensitive to the dynamical models.

Renormalons at low x

The role of infrared and ultraviolet renormalons are discussed in context of the leading log (1/ x) approximation of perturbative QCD. We generalize the BFKL equation for the case of running QCD coupling constant and show that the uncertainties related to the contribution of infrared renormalons turn out to be smaller, than the shadowing correction to the total cross section, in the region of small x. The contributions of infrared and ultraviolet renormalons to the BFKL equation are studied. It is shown that infrared renormalons generate the nonperturbative contribution of the order of √ A 2/ Q 2, which does not appear in the Wilson Operator Product Expansion. The solution of the BFKL equation with a running coupling constant is discussed.

Analytic properties of the QCD running coupling constant and τ decay

A non-perturbative expansion method which gives a well-defined analytic continuation of the running coupling constant from the spacelike to the timelike region is applied to the inclusive semileptonic decay of the τ-lepton. The method allows us to evaluate R τ by integration over the non-perturbative physical region, rather than by using Cauchy's theorem, and hence to incorporate threshold effects in a very direct way. Within our framework the difference between the effective coupling constants in the timelike and spacelike domains can be substantial and is not simply a matter of the standard π 2 terms.

High-colour pomerons with a running coupling constant in the Hartree-Fock approximation

The Hartree-Fock approximation is applied to study the “high-colour pomerons” in the system of many reggeized gluons with a running QCD coupling constant. It is shown that, contrary to the fixed coupling case, the high-colour pomerons result in being supercritical, although with a smaller intercept than the multipomeron states.

QCD running coupling constant in the timelike region

By using a non-perturbative expansion and the dispersion relation for the Adler D-function we propose a new method for constructing the QCD effective coupling constant in the timelike region.

The relevant scale parameter in the high temperature phase of QCD

We introduce the running coupling constant of QCD in the high temperature phase, g 2(T) , through a renormalization scheme where the dimensional reduction is optimal at the one-loop level. We then calculate the relevant scale parameter, Λ T , which characterizes the running of g 2(T) with T, using the background field method in the static sector. It is found that Λ T/Λ=e( γE+ 1 22 )/(4π) ≈ 0.148 . We further verify that the coupling g 2(T) is also optimal for lattice perturbative calculations. Our result naturally explains why the high temperature limit of QCD sets in at temperatures as low as a few times the critical temperature. In addition, our Λ t agrees remarkably well with the scale parameter determined from the lattice measurement of the spatial string tension of the SU(2) gauge theory at high T.

Is there a vector boson coupling to baryon number?

Discrepancies in measurements of the QCD running coupling constant are consistent with a new vector gauge boson coupling to baryon number with strength β B ∼ 0.1. Upsilon decays constrain such a boson to have a mass greater than about 35 GeV. It is difficult to observe such a boson in Z decays, e +e − annihilation, or hadronic interactions.

Reggeized gluons with a running coupling constant

The equation for two reggeized gluons in the vacuum channel is generalized to take into account the running QCD coupling constant on the basis of the bootstrap condition for gluon reggeization. Both the gluon trajectory as a function of momentum and the interaction as a function of distance grow like log log in the ultraviolet. The resulting equation depends on the confinement region. With a simple parametrization of its influence by an effective gluon mass the pomeron intercept turns out to be much smaller than for a fixed coupling constant (the BFKL pomeron).

Pomeron with a running coupling constant: intercept and slope

The equation for two reggeized gluons with running QCD coupling constant, proposed earlier on the basis of the bootstrap condition, is investigated by the variational technique in the vacuum channel. The pomeron intercept and slope are calculated as a function of the infrared cutoff parameter m (“the gluon mass”). The intercept depends weakly on m staying in the region 0.3–0.5 for values of m in the range 0.3–1.0 GeV. The slope goes down from 5.0 GeV −2 to 0.08 GeV −2 in this range. The optimal value of m seems to lie in the region 0.6–0.8 GeV with the intercept 0.35 and slope 0.35-0.15. The character of the pomeron singularity: pole or cut, is also discussed.

2- and 3-point gluon correlation functions on the lattice

I present some preliminary results, obtained in collaboration with C. Bernard and A. Soni, for the lattice evaluation of 2- and 3-point gluon correlation functions in momentum space, with emphasis on the amputated 3-gluon vertex function. The final goal of this approach is the study of the running QCD coupling constant as defined from the amputated 3-gluon vertex.

QCD analysis of the tau hadronic width

The total τ hadronic width can be accurately calculated using analyticity and the operator product expansion. The theoretical analysis of this observable is updated to include all available perturbative and non-perturbative corrections. Experimental measurements of τ decay rates are used to determine with high precision the QCD running coupling constant at the scale of the τ mass. The analysis is also used to study the present discrepancy between the experimental measurements of the leptonic branching fractions of the τ and its total lifetime.

Renormalization scheme dependence in the next-to-next-to-leading order and the high-energy behaviour in QCD

It is pointed out that the renormalization scheme dependence of the next-to-next-to-leading order predictions of observable quantities in QCD is very strong. It is shown that changing continuously the scheme parameters we may formally obtain qualitatively different results. As an illustration the QCD contribution to theRe+e− is considered. The strong renormalization scheme dependence of the QCD running coupling constant is also explicity demonstrated.

The chromodielectric model: Confinement, chiral symmetry breaking, and the pion

The chromodielectric model is a chirally symmetric effective model of quantum chromodynamics, where the long-range parts of the (nonabelian) gluon self-interactions are assumed to give rise to a color-dielectric function. We study dynamical chiral symmetry breaking in this model using a local approximation, where the Goldstone pion is assumed to be essentially point-like. The usual running coupling constant of QCD is included to describe the high-momentum region. We show how confinement and the Goldstone pion arise naturally and simultaneously in this approach and also that familiar, successful chiral bag models are obtained as a limiting case.

Color confinement and the quantum-chromodynamic vacuum. II. Gluon propagation and the Coulomb interaction.

In an earlier paper the color-confining interaction was identified as the vacuum expectation value of the modified Coulomb operator. This vacuum expectation value is related first to the QCD running coupling constant and then to the gluon propagator in the physical vacuum. If the propagator is less singular at p=0 than predicted from perturbation theory, the required infrared singularity in the interaction arises naturally. Nonperturbative solutions for both the running coupling constant and the modified Coulomb interaction are presented.

Properties of quark matter governed by quantum chromodynamics (I). The screened charge

A genuine screened charge (coupling) is constructed for quark matter, subject to a flavour constraint, governed by quantum chromodynamics. This is achieved by (i) working in the Coulomb gauge and (ii) using an unambigous renormalization prescription (formulated in terms of the Lie differential equations) to overcome the problems of the previous attempts. Since we must standardize with the vacuum QCD running (experimental) coupling constant at some asymptotic spacelike momentum transfer, | p 0| , it is ensured that the matter contribution is negligible at this point by the condition, μ⪡| p 0| , where μ is the matter (flavour) chemical potential. The screened charge (coupling) has the expected property of merging with the vacuum QCD running coupling constant in the ultraviolet ( p 2 → ∞) and going to zero in the infrared ( p 2 → 0) , indicative of complete screening above a critical value of μ (or density) = μ c . The critical length scale at which the screened charge diverges is well below the vacuum QCD scale length, Λ, and screening always renders the screened charge smaller than its vacuum QCD counterpart. This gives greater access to the long-distance phenomena besides providing a better (see paper II) expansion parameter for perturbative calculations. The richer physical content of the screened charge is pointed out. The quark mass is included and taken to be renormalized on-shell.

Properties of quark matter governed by quantum chromodynamics (II). Renormalization schemes in quark matter

Renormalization schemes are examined (in the Coulomb gauge) for quantum chromodynamics in the presence of quark matter. We demand that the effective coupling constant for all schemes become congruent with the vacuum QCD running coupling constant as the matter chemical potential, μ, goes to zero. Also, to enable us to standardize with the vacuum QCD running coupling constant at some asymptotic momentum transfer, | p 0|, we keep μ ⪡ ¦ p 0¦ , to ensure that the matter contribution is negligible at this point. This means all schemes merge with vacuum QCD at | p 0| and beyond. Two renormalization group invariants are shown to emerge: (i) the effective or invariant charge, g inv 2, which is, however, scheme dependent and (ii) g 2( M)/ S( M), where S( M) −1 is the Coulomb propagator, which is scheme independent. The only scheme in which g inv 2 is scheme independent and identical to g 2( M)/ S( M) is the screened charged scheme (previous paper) characterised by the normalization of the entire Green function, S −1, to unity. We conclude that this is the scheme to be used if one wants to identify with the experimental effective coupling in perturbation theory. However, if we do not restrict to perturbation theory all schemes should be allowed. Although we discuss matter QCD in the Coulomb gauge, the above considerations are quite general to gauge theories in the presence of matter.

αS( KT2) versus αS( K2) in QCD jet development: Some quantitative remarks

Properties of QCD jets for two proposed choices of the variable appearing in the QCD running coupling constant, α S ( K 2) or α S ( K T 2) with K T 2= z(1− z) K 2, have been studied in detail by the use of Monte Carlo techniques. Contrary to expectations based on approximate analytic calculations it is found that when substituting K 2 with K T 2 in α S the growth with energy of the multiplicity of the quanta is not sensibly altered, and it is even slowed down for extreme values of the non-perturbative cutoff, which delimits the minimum virtual mass of the quanta. It appears, therefore, that the analysis of subleading effects may be totally misleading if phase-space constraints are not taken into account exactly, which is possible to do in Monte Carlo calculations.