Beta-function Coefficient Research Articles

Anomalous U(1)'s and proton stability in brane models

We consider the most general generation-independent U(1) gauge symmetry consistent with the presence of Yukawa couplings for all quarks and leptons in the SUSY version of the Standard Model. This U(1) has generically mixed anomalies with SM groups, which cannot be cancelled by the Green-Schwarz mechanism of heterotic D=4 strings. We argue that these anomalies can in principle be cancelled by the generalized Green-Schwarz mechanism present in field theories corresponding to D-branes at singularities. Moreover, unlike the heterotic case, once the U(1) symmetry is broken it may remain as an exact perturbative global symmetry in the low energy theory. Applying this scheme to the SUSY SM we find that gauging such a general U(1):1) B and L violating operators at least up to dim=3,4,5,6 are generically forbidden; 2)The $\mu$-term is generically supressed. We also study the properties of a U(1) symmetry whose mixed anomalies with the different SM gauge groups are in the ratio of the beta function coefficients $\beta_a$. This relation has been shown to hold in certain orientifold models. In all cases the U(1) remains as a global symmetry at the orientifold singularity, the SM Higgs can break it at the electroweak scale, making possible to relate the blowing-up of the singularity with electroweak symmetry breaking.

The uncertainty in αs ( MZ2) determined from hadronic tau decay measurements

We show that QCD Minkowski observables such as the e + e − R-ratio and the hadronic tau decay R τ are completely determined by the effective charge (EC) beta-function, ϱ( x), corresponding to the Euclidean QCD vacuum polarization Adler D-function, together with the next-to-leading order (NLO) perturbative coefficient of D. An efficient numerical algorithm is given for evaluating R, R τ from a weighted contour integration of D ( se iΘ around a circle in the complex squared energy s-plane, with ϱ( x) used to evolve in s around the contour. The EC beta-function can be truncated at next-to-NLO (NNLO) using the known exact perturbative calculation or the uncalculated N 3 LO and higher terms can be approximated by the portion containing the highest power of b, the first QCD beta-function coefficient. The difference between the R, R τ constructed using the NNLO and “leading- b” resummed versions of ϱ( x) provides an estimate of the uncertainty due to the uncalculated higher order corrections. Simple numerical parametrizations are given to facilitate these fits. For R τ we estimate an uncertainty Δα s ( m τ 2) ≅ 0.01, corresponding to Δα s ( M z 2) ≅ 0.002. Thus there is no necessary implication that the uncalculated higher order corrections contribute a large uncertainty in α s ( M Z 2) as had been claimed by other authors based on analogous all-orders resummations. We show that these earlier estimates are dependent on the renormalization scheme chosen, a difficulty which is avoided in our formulation.

Gauge unification and dynamical supersymmetry breaking

Under the assumption that all the gauge groups in supersymmetric theories unify at the fundamental scale, the numbers and the mass scales of messenger quarks and leptons, as well as the beta-function coefficient of the sector for dynamical supersymmetry breaking are constrained depending on various gauge mediation mechanisms. For this, we use one-loop renormalization group equations and draw constraints on the scales in each gauge mediation model.

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.

RS-invariant all-orders renormalon resummations for some QCD observables

We propose a renormalon-inspired resummation of QCD perturbation theory based on approximating the renormalization scheme (RS)-invariant effective charge beta-function coefficients by the portion containing the highest power of b= 1 6 (11N−2N f) for SU( N) QCD with N f quark flavours. This can be accomplished using exact large- N f all-orders results. The resulting resummation is RS invariant and the exact next-to-leading order (NLO) and next-to-NLO (NNLO) coefficients in any RS are included. This improves on a previously employed naive resummation of the leading- b piece of the perturbative coefficients which is RS dependent, making its comparison with fixed-order perturbative results ambiguous. The RS-invariant resummation is used to assess the reliability of fixed-order perturbation theory for the e + e − R-ratio, the analogous τ-lepton decay ratio R τ and deep inelastic scattering (DIS) sum rules, by comparing it with the exact NNLO results in the effective charge RS. For the R-ratio and R,, where large-order perturbative behaviour is dominated by a leading ultra-violet renormalon singularity, the comparison indicates fixed-order perturbation theory to be very reliable. For DIS sum rules, which have a leading infra-red renormalon singularity, the performance is rather poor. In this way we estimate that at LEP/ SLD energies ideal data on the R-ratio could determine α s ( M z ) to three-significant figures, and for the R τ we estimate a theoretical uncertainty δα s ( m τ ) ≅ 0.008 corresponding to δα s ( M z ) ≅ 0.001 This encouragingly small uncertainty is much less than has recently been deduced from comparison with the ambiguous naive resummation.