Perturbative QCD in acceptable schemes with holomorphic coupling

Abstract

Perturbative QCD in mass independent schemes leads in general to running coupling $a(Q^2)$ which is nonanalytic (nonholomorphic) in the regime of low spacelike momenta $|Q^2| \lesssim 1 \ {\rm GeV}^2$. Such (Landau) singularities are inconvenient in the following sense: evaluations of spacelike physical quantities ${\cal D}(Q^2)$ with such a running coupling $a(\kappa Q^2)$ ($\kappa \sim 1$) give us expressions with the same kind of singularities, while the general principles of local quantum field theory require that the mentioned physical quantities have no such singularities. In a previous work, certain classes of perturbative mass independent beta functions were found such that the resulting coupling was holomorphic. However, the resulting perturbation series showed explosive increase of coefficients already at ${\rm N}^4{\rm LO}$ order, as a consequence of the requirement that the theory reproduce the correct value of the $\tau$ lepton semihadronic strangeless decay ratio $r_{\tau}$. In this work we successfully extend the construction to specific classes of perturbative beta functions such that the perturbation series do not show explosive increase of coefficients, the perturbative coupling is holomorphic, and the correct value of $r_{\tau}$ is reproduced. In addition, we extract, with Borel sum rule analysis of the V+A channel of the semihadronic strangeless decays of $\tau$ lepton, reasonable values of the corresponding D=4 and D=6 condensates.Comment: v3: 27 pages; 12 figures; includes new Sec.V - sum rule analysis for semihadronic decays of tau lepton in a proposed QCD scheme; new Refs.: [114,115,118-132]; to appear in Int.J.Mod.Phys.

Nearly perturbative lattice-motivated QCD coupling with zero IR limit

“…The coupling is constructed in a dispersive way, resulting as a byproduct in the holomorphic behavior of A(Q 2 ) in the complex Q 2 -plane which reflects the holomorphic behavior of the spacelike QCD observables. Application of the Borel sum rules to τ -decay V + A spectral functions allows us to obtain values for the gluon (dimension-4) condensate and the dimension-6 condensate, which reproduce the measured OPAL and ALEPH data to a significantly better precision than the perturbative MS coupling approach.3 It is possible to show that pQCD renormalization schemes exist in which pQCD coupling a(Q 2 ) is holomorphic for Q 2 ∈ C\(−∞, −M 2 thr ] and at the same time reproduces the high-energy QCD phenomenology as well as the semihadronic τ -lepton decay physics [23][24][25]. 4 MiniMOM scheme is known at present to four loops [18][19][20].…”

“…If in Eq. (25) we increased the power index, i.e., N max > 5, the numerical results for A(Q 2 ) would change insignificantly: A would merge even slightly better with a in the UV regime, but the number of conditions and parameters in A would increase. On the other hand, N max = 5 is sufficiently high for the application of OPE.…”

“…As always, the pion peak term 2π 2 f 2 π δ(σ − m 2 π ) (where f π = 0.1305 GeV) is included in ω exp (σ); this accounts for the pion contribution, but nonetheless does not include in r τ,exp the main part of the mass effects m π = 0, i.e., ∆r τ (m u,d = 0); this is then consistent with the nonpresence of the mass effects m u,d = 0 in the theoretical expression (28) for r τ and r (D=0) τ . 25 Further, (1.198 ± 0.006) is the value of the above integral over σ, and 0.005 represents the subtraction of the D = 6 term with O 6 V +A = +0.0013 GeV 6 , cf. Eq.…”

Lattice-motivated holomorphic nearly perturbative QCD

“…A of [72]], giving in this scheme the value a = 0.28043/π at Q 2 = (2m c ) 2 and N f = 3, and the Lambert scale value Λ L = 1.153 GeV. For more details on this procedure, we refer to [37,48,62,73]. We point out that we construct our coupling A(Q 2 ) in (Lambert-)MiniMOM renormalization scheme, and not in the usual MS scheme, in order to make the comparison of Fig.…”

How to perform a QCD analysis of DIS in analytic perturbation theory

“…For further literature on various analytic QCD models, we refer to review articles [10,20,21]. Some newer constructions of analytic models in QCD of A 1 (Q 2 ) include those based on specific classes of β functions with nonperturbative contributions [22] or without such contributions [23][24][25] and those based on modifications of the the spectral density ρ…”