Nearly perturbative lattice-motivated QCD coupling with zero IR limit

Abstract

The product of the gluon dressing function and the square of the ghost dressing function in the Landau gauge can be regarded to represent, apart from the inverse power corrections 1/Q 2n , a nonperturbative generalization A(Q 2 ) of the perturbative QCD running coupling a(Q 2 ) (≡ αs(Q 2 )/π). Recent large volume lattice calculations for these dressing functions indicate that the coupling defined in such a way goes to zero as A(Q 2 ) ∼ Q 2 when the squared momenta Q 2 go to zero (Q 2 1 GeV 2 ). In this work we construct such a QCD coupling A(Q 2 ) which fulfills also various other physically motivated conditions. At high momenta it becomes the underlying perturbative coupling a(Q 2 ) to a very high precision. And at intermediate low squared momenta Q 2 ∼ 1 GeV 2 it gives results consistent with the data of the semihadronic τ lepton decays as measured by OPAL and ALEPH. 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]. 5 In this scheme, however, we rescale Q 2 from the Λ MM to the usual Λ MS convention. 6 In Ref. [29], the matching of A(Q 2 ) and dA(Q 2 )/d ln Q 2 at an IR/UV transition scale Q 2 0 ∼ 1 GeV 2 is imposed, fixing the values of A(0) > 0 and Q 2 0 . On the other hand, our coupling A(Q 2 ) will be holomorphic, no explicit IR/UV matching scale will exist. Instead of the matching, we will impose various physically motivated conditions which will affect simultaneously the behavior of A(Q 2 ) in the UV and IR regimes.10 In principle, we could construct A in any other scheme, e.g., in MS scheme, but then it would not be clear how such a coupling compares with A latt of Ref.[32] in the deep IR regime. For an application and discussion of the MiniMOM scheme in pQCD, see Ref.[56].

Renormalon-motivated evaluation of QCD observables

“…In practice, this ansatz is made in a specific renormalization scheme, the Lambert MiniMOM (LMM) [34], because in that scheme the IR-safe (and holomorphic) QCD coupling was constructed a(Q 2 ) → A(Q 2 ) [16], which at high Q 2 practically coincides with the underlying pQCD coupling a(Q 2 ) (in LMM), reproduces the correct semihadronic τ -decay ratio r τ ≈ 0.20, and behaves as A(Q 2 ) ∼ Q 2 when Q 2 → 0 as suggested by large-volume lattice data on gluon and ghost propagator dressing functions in the Landau gauge [13][14][15]. This QCD variant is called 3δ AQCD, because the spectral (discontinuity) function ρ A (σ) ≡ ImA(Q 2 = −σ − i ) in the low-σ regime (0 ≤ σ 1 GeV 2 ) is parametrized by three Dirac-delta functions, while ρ A (σ) for higher σ coincides with its underlying pQCD version ρ a (σ).…”

“…We are interested in the Adler function in this c 2 = −4.9 Lambert scheme, because in this scheme an IR-safe (and holomorphic) QCD coupling A(Q 2 ) was constructed [17], which at high Q 2 practically coincides with the underlying pQCD coupling a(Q 2 ) and reproduces the correct r τ ≈ 0.20; however, at Q 2 → 0 the coupling is nonzero, 0 < A(0) < ∞, in contrast with the aforementioned 3δ AQCD coupling [16]. This QCD variant is called 2δ AQCD, because its spectral function ρ A (σ) ≡ ImA(Q 2 = −σ−i ) in the low-σ regime is parametrized by two Diracdelta functions.…”

“…Several versions of QCD holomorphic couplings have been applied in evaluations of various QCD quantities [24][25][26][27]. [37] Two recently constructed QCD variants with holomorphic couplings A(Q 2 ), the aforementioned 2δ AQCD [17] and 3δ AQCD [16], fulfill several phenomenological constraints of the low-Q 2 QCD (|Q 2 | 1 GeV 2 ) as mentioned earlier. The integrals in Eq.…”

Renormalization scheme and gauge (in)dependence of the generalized Crewther relation: what are the real grounds of the β-factorization property?

“…[70]. It is widely used in different QCDoriented studies [49,[93][94][95][96][97][98][99][100]. In this section we will use this mMOM-scheme in the SU(N c ) theory to find out the constraints imposed by its gauge-dependence on the conditions of existence of the fundamental property of the β-function factorization in the GCR.…”

Bjorken polarized sum rule and infrared-safe QCD couplings

“…In this work we fit the theoretical OPE expressions to the experimental BSR results in pQCD, in (F)APT, and two additional extensions of QCD to low Q 2 , namely the 2δ [22,23] and 3δ [24,25] AQCD. The latter two extensions have the coupling A(Q 2 ) [the analog of the pQCD coupling a(Q 2 )] which is free of Landau singularities and physically motivated in the entire relevant regime of Q 2 in the complex plane, Q 2 ∈ C\(−∞, −M 2 thr ], where M 2 thr 1 GeV 2 is a positive threshold scale.…”